docs/htmldoc/mathcomp.field.finfield.html


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<h1 class="libtitle">Library mathcomp.field.finfield</h1>

<div class="code">
<span class="comment">(*&nbsp;(c)&nbsp;Copyright&nbsp;2006-2016&nbsp;Microsoft&nbsp;Corporation&nbsp;and&nbsp;Inria.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
&nbsp;Distributed&nbsp;under&nbsp;the&nbsp;terms&nbsp;of&nbsp;CeCILL-B.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*)</span><br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/>

<br/>
</div>

<div class="doc">
  Additional constructions and results on finite fields.                    

<div class="paragraph"> </div>

         FinFieldExtType L == A FinFieldType structure on the carrier of L, 
                              where L IS a fieldExtType F structure for an  
                              F that has a finFieldType structure. This     
                              does not take any existing finType structure  
                              on L; this should not be made canonical.      
 FinSplittingFieldType F L == A SplittingFieldType F structure on the       
                              carrier of L, where L IS a fieldExtType F for 
                              an F with a finFieldType structure; this      
                              should not be made canonical.                 
          Import FinVector :: Declares canonical default finType, finRing,  
                              etc structures (including FinFieldExtType     
                              above) for abstract vectType, FalgType and    
                              fieldExtType over a finFieldType. This should 
                              be used with caution (e.g., local to a proof) 
                              as the finType so obtained may clash with the 
                              canonical one for standard types like matrix. 
      PrimeCharType charRp == The carrier of a ringType R such that         
                              charRp : p \in [char R] holds. This type has  
                              canonical ringType, ..., fieldType structures 
                              compatible with those of R, as well as        
                              canonical lmodType 'F_p, ..., algType 'F_p    
                              structures, plus an FalgType structure if R   
                              is a finUnitRingType and a splittingFieldType 
                              struture if R is a finFieldType.              
 FinSplittingFieldFor nz_p == sigma-pair whose sval is a splittingFieldType 
                              that is the splitting field for p : {poly F}  
                              over F : finFieldType, given nz_p : p != 0.   
 PrimePowerField pr_p k_gt0 == sigma2-triple whose s2val is a finFieldType  
                              of characteristic p and order m = p ^ k,      
                              given pr_p : prime p and k_gt0 : k &gt; 0.       
   FinDomainFieldType domR == A finFieldType structure on a finUnitRingType 
                              R, given domR : GRing.IntegralDomain.axiom R. 
                              This is intended to be used inside proofs,    
                              where one cannot declare Canonical instances. 
                              Otherwise one should construct explicitly the 
                              intermediate structures using the ssralg and  
                              finalg constructors, and finDomain_mulrC domR 
                              finDomain_fieldP domR to prove commutativity  
                              and field axioms (the former is Wedderburn's  
                              little theorem).                              
 FinDomainSplittingFieldType domR charRp == A splittingFieldType structure  
                              that repackages the two constructions above.   
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>

<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span> <span class="id" title="var">GRing.Theory</span> <span class="id" title="var">FinRing.Theory</span>.<br/>
<span class="id" title="keyword">Local Open</span> <span class="id" title="keyword">Scope</span> <span class="id" title="var">ring_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinRing"><span class="id" title="section">FinRing</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinRing.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Ring.Exports.finRingType"><span class="id" title="abbreviation">finRingType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finRing_nontrivial"><span class="id" title="lemma">finRing_nontrivial</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinRing.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 1%<span class="id" title="var">g</span>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="finRing_gt1"><span class="id" title="lemma">finRing_gt1</span></a> : 1 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinRing.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinRing"><span class="id" title="section">FinRing</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinField"><span class="id" title="section">FinField</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_finField_unit"><span class="id" title="lemma">card_finField_unit</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">unit</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f7c6b2be51cd10aae4ae8951352903f1"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="finField_unit"><span class="id" title="definition">finField_unit</span></a> <span class="id" title="var">x</span> (<span class="id" title="var">nz_x</span> : <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#FinRing.unit"><span class="id" title="abbreviation">FinRing.unit</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#etrans"><span class="id" title="definition">etrans</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.unitfE"><span class="id" title="definition">unitfE</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.field.finfield.html#nz_x"><span class="id" title="variable">nz_x</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="expf_card"><span class="id" title="lemma">expf_card</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finField_genPoly"><span class="id" title="lemma">finField_genPoly</span></a> : <a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">X</span></a><a class="idref" href="mathcomp.algebra.poly.html#9f0d1035fe3072a93b6e6065c1932def"><span class="id" title="notation">^</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#24846b5795605f82696a43aa191874ea"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#24846b5795605f82696a43aa191874ea"><span class="id" title="notation">prod_x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#24846b5795605f82696a43aa191874ea"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.poly.html#ffd3fc7e3c529f4febe87040923e7332"><span class="id" title="notation">X</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">P</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#24846b5795605f82696a43aa191874ea"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finCharP"><span class="id" title="lemma">finCharP</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">p</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finField_is_abelem"><span class="id" title="lemma">finField_is_abelem</span></a> : <a class="idref" href="mathcomp.solvable.abelian.html#is_abelem"><span class="id" title="definition">is_abelem</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_finCharP"><span class="id" title="lemma">card_finCharP</span></a> <span class="id" title="var">p</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.field.finfield.html#n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinField"><span class="id" title="section">FinField</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="CardVspace"><span class="id" title="section">CardVspace</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="CardVspace.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>) (<a name="CardVspace.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="CardVspace.Vector"><span class="id" title="section">Vector</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="CardVspace.Vector.cvT"><span class="id" title="variable">cvT</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.class_of"><span class="id" title="record">Vector.class_of</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.T"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Let</span> <a name="CardVspace.Vector.vT"><span class="id" title="variable">vT</span></a> := <a class="idref" href="mathcomp.algebra.vector.html#Vector.Pack"><span class="id" title="constructor">Vector.Pack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a>) <a class="idref" href="mathcomp.field.finfield.html#CardVspace.Vector.cvT"><span class="id" title="variable">cvT</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_vspace"><span class="id" title="lemma">card_vspace</span></a> (<span class="id" title="var">V</span> : <a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">vspace</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.Vector.vT"><span class="id" title="variable">vT</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#V"><span class="id" title="variable">V</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">dim</span></a> <a class="idref" href="mathcomp.field.finfield.html#V"><span class="id" title="variable">V</span></a>)%<span class="id" title="var">N</span>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_vspacef"><span class="id" title="lemma">card_vspacef</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">{:</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.Vector.vT"><span class="id" title="variable">vT</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">}</span></a>%<span class="id" title="var">VS</span><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#CardVspace.T"><span class="id" title="variable">T</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.Vector"><span class="id" title="section">Vector</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="CardVspace.caT"><span class="id" title="variable">caT</span></a> : <a class="idref" href="mathcomp.field.falgebra.html#Falgebra.class_of"><span class="id" title="record">Falgebra.class_of</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.T"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Let</span> <a name="CardVspace.aT"><span class="id" title="variable">aT</span></a> := <a class="idref" href="mathcomp.field.falgebra.html#Falgebra.Pack"><span class="id" title="constructor">Falgebra.Pack</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a>) <a class="idref" href="mathcomp.field.finfield.html#CardVspace.caT"><span class="id" title="variable">caT</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_vspace1"><span class="id" title="lemma">card_vspace1</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|(</span></a>1%<span class="id" title="var">VS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">vspace</span></a> <a class="idref" href="mathcomp.field.finfield.html#CardVspace.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">)|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#CardVspace.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#CardVspace"><span class="id" title="section">CardVspace</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="VectFinMixin"><span class="id" title="lemma">VectFinMixin</span></a> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Ring.Exports.finRingType"><span class="id" title="abbreviation">finRingType</span></a>) (<span class="id" title="var">vT</span> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.vectType"><span class="id" title="abbreviation">vectType</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.mixin_of"><span class="id" title="record">Finite.mixin_of</span></a> <a class="idref" href="mathcomp.field.finfield.html#vT"><span class="id" title="variable">vT</span></a>.<br/>

<br/>
</div>

<div class="doc">
 These instancces are not exported by default because they conflict with    
 existing finType instances such as matrix_finType or primeChar_finType.     
</div>
<div class="code">
<span class="id" title="keyword">Module</span> <a name="FinVector"><span class="id" title="module">FinVector</span></a>.<br/>
<span class="id" title="keyword">Section</span> <a name="FinVector.Interfaces"><span class="id" title="section">Interfaces</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinVector.Interfaces.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">vT</span> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.vectType"><span class="id" title="abbreviation">vectType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinVector.Interfaces.F"><span class="id" title="variable">F</span></a>) (<span class="id" title="var">aT</span> : <a class="idref" href="mathcomp.field.falgebra.html#Falgebra.Exports.FalgType"><span class="id" title="abbreviation">FalgType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinVector.Interfaces.F"><span class="id" title="variable">F</span></a>) (<span class="id" title="var">fT</span> : <a class="idref" href="mathcomp.field.fieldext.html#FieldExt.Exports.fieldExtType"><span class="id" title="abbreviation">fieldExtType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinVector.Interfaces.F"><span class="id" title="variable">F</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">vect_finType</span> <span class="id" title="var">vT</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> <a class="idref" href="mathcomp.field.finfield.html#vT"><span class="id" title="variable">vT</span></a> (<a class="idref" href="mathcomp.field.finfield.html#VectFinMixin"><span class="id" title="lemma">VectFinMixin</span></a> <a class="idref" href="mathcomp.field.finfield.html#vT"><span class="id" title="variable">vT</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Falg_finType</span> <span class="id" title="var">aT</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> <a class="idref" href="mathcomp.field.finfield.html#aT"><span class="id" title="variable">aT</span></a> (<a class="idref" href="mathcomp.field.finfield.html#VectFinMixin"><span class="id" title="lemma">VectFinMixin</span></a> <a class="idref" href="mathcomp.field.finfield.html#aT"><span class="id" title="variable">aT</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldExt_finType</span> <span class="id" title="var">fT</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> <a class="idref" href="mathcomp.field.finfield.html#fT"><span class="id" title="variable">fT</span></a> (<a class="idref" href="mathcomp.field.finfield.html#VectFinMixin"><span class="id" title="lemma">VectFinMixin</span></a> <a class="idref" href="mathcomp.field.finfield.html#fT"><span class="id" title="variable">fT</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">Falg_finRingType</span> <span class="id" title="var">aT</span> := <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">finRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldExt_finRingType</span> <span class="id" title="var">fT</span> := <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">finRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#fT"><span class="id" title="variable">fT</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fieldExt_finFieldType</span> <span class="id" title="var">fT</span> := <a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">finFieldType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#fT"><span class="id" title="variable">fT</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="FinVector.finField_splittingField_axiom"><span class="id" title="lemma">finField_splittingField_axiom</span></a> <span class="id" title="var">fT</span> : <a class="idref" href="mathcomp.field.galois.html#SplittingField.axiom"><span class="id" title="definition">SplittingField.axiom</span></a> <a class="idref" href="mathcomp.field.finfield.html#fT"><span class="id" title="variable">fT</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinVector.Interfaces"><span class="id" title="section">Interfaces</span></a>.<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinVector"><span class="id" title="module">FinVector</span></a>.<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="FinFieldExtType"><span class="id" title="abbreviation">FinFieldExtType</span></a> := <a class="idref" href="mathcomp.field.finfield.html#fieldExt_finFieldType"><span class="id" title="definition">FinVector.fieldExt_finFieldType</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="FinSplittingFieldAxiom"><span class="id" title="abbreviation">FinSplittingFieldAxiom</span></a> := (<a class="idref" href="mathcomp.field.finfield.html#finField_splittingField_axiom"><span class="id" title="lemma">FinVector.finField_splittingField_axiom</span></a> <span class="id" title="var">_</span>).<br/>
<span class="id" title="keyword">Notation</span> <a name="FinSplittingFieldType"><span class="id" title="abbreviation">FinSplittingFieldType</span></a> <span class="id" title="var">F</span> <span class="id" title="var">L</span> :=<br/>
&nbsp;&nbsp;(<a class="idref" href="mathcomp.field.galois.html#SplittingField.Exports.SplittingFieldType"><span class="id" title="abbreviation">SplittingFieldType</span></a> <span class="id" title="var">F</span> <span class="id" title="var">L</span> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingFieldAxiom"><span class="id" title="abbreviation">FinSplittingFieldAxiom</span></a>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="PrimeChar"><span class="id" title="section">PrimeChar</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="PrimeChar.p"><span class="id" title="variable">p</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="PrimeChar.PrimeCharRing"><span class="id" title="section">PrimeCharRing</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="PrimeChar.PrimeCharRing.R0"><span class="id" title="variable">R0</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Ring.Exports.ringType"><span class="id" title="abbreviation">ringType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="PrimeCharType"><span class="id" title="definition">PrimeCharType</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.PrimeCharRing.R0"><span class="id" title="variable">R0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#predArgType"><span class="id" title="definition">predArgType</span></a> := <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.PrimeCharRing.R0"><span class="id" title="variable">R0</span></a>.<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="PrimeChar.PrimeCharRing.charRp"><span class="id" title="variable">charRp</span></a> : <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.PrimeCharRing.R0"><span class="id" title="variable">R0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_eqType</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">eqType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#cb062fd562aed512787df99359c6e3f2"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_choiceType</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">choiceType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_zmodType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#af6385fc2df84aeeec6855073f75cc68"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af6385fc2df84aeeec6855073f75cc68"><span class="id" title="notation">zmodType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#af6385fc2df84aeeec6855073f75cc68"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af6385fc2df84aeeec6855073f75cc68"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_ringType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#dee4f3431027813095272c568fc6b5ce"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#dee4f3431027813095272c568fc6b5ce"><span class="id" title="notation">ringType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#dee4f3431027813095272c568fc6b5ce"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#dee4f3431027813095272c568fc6b5ce"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="primeChar_scale"><span class="id" title="definition">primeChar_scale</span></a> <span class="id" title="var">a</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a>.<br/>

<br/>
<span class="id" title="keyword">Let</span> <a name="PrimeChar.PrimeCharRing.natrFp"><span class="id" title="variable">natrFp</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#inZp"><span class="id" title="definition">inZp</span></a> <a class="idref" href="mathcomp.field.finfield.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">)%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">%:</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#af5c1d7e13410a0a6c3dff5441ac8477"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scaleA"><span class="id" title="lemma">primeChar_scaleA</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">p</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.finfield.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">p</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#22058a36a53dac65c94ca403bc62650a"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.field.finfield.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">×</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">p</span></a><a class="idref" href="mathcomp.field.finfield.html#5f6f59b8095d2eaa0b5a55ed9129580f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scale1"><span class="id" title="lemma">primeChar_scale1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.field.finfield.html#primeChar_scale"><span class="id" title="definition">primeChar_scale</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scaleDr"><span class="id" title="lemma">primeChar_scaleDr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_distributive"><span class="id" title="definition">right_distributive</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scale"><span class="id" title="definition">primeChar_scale</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#327bb2f0da6fd7c01a004dedcfc2dee4"><span class="id" title="notation">R</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scaleDl"><span class="id" title="lemma">primeChar_scaleDl</span></a> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scale"><span class="id" title="definition">primeChar_scale</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#ae4d81913e6239182a9ac7467ffde8cd"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.field.finfield.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Definition</span> <a name="primeChar_lmodMixin"><span class="id" title="definition">primeChar_lmodMixin</span></a> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodMixin"><span class="id" title="abbreviation">LmodMixin</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scaleA"><span class="id" title="lemma">primeChar_scaleA</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scale1"><span class="id" title="lemma">primeChar_scale1</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.field.finfield.html#primeChar_scaleDr"><span class="id" title="lemma">primeChar_scaleDr</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scaleDl"><span class="id" title="lemma">primeChar_scaleDl</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_lmodType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lmodule.Exports.LmodType"><span class="id" title="abbreviation">LmodType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_lmodMixin"><span class="id" title="definition">primeChar_lmodMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scaleAl"><span class="id" title="lemma">primeChar_scaleAl</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lalgebra.axiom"><span class="id" title="definition">GRing.Lalgebra.axiom</span></a> ( <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a>).<br/>
 <span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_LalgType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Lalgebra.Exports.LalgType"><span class="id" title="abbreviation">LalgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scaleAl"><span class="id" title="lemma">primeChar_scaleAl</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_scaleAr"><span class="id" title="lemma">primeChar_scaleAr</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Algebra.axiom"><span class="id" title="definition">GRing.Algebra.axiom</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_LalgType"><span class="id" title="definition">primeChar_LalgType</span></a>.<br/>
 <span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_algType</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Algebra.Exports.AlgType"><span class="id" title="abbreviation">AlgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_scaleAr"><span class="id" title="lemma">primeChar_scaleAr</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.PrimeCharRing"><span class="id" title="section">PrimeCharRing</span></a>.<br/>

<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_unitRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#f02859ca87d7563e473a6ba817bdc33f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f02859ca87d7563e473a6ba817bdc33f"><span class="id" title="notation">unitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f02859ca87d7563e473a6ba817bdc33f"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f02859ca87d7563e473a6ba817bdc33f"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_unitAlgType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.UnitRing.Exports.unitRingType"><span class="id" title="abbreviation">unitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">unitAlgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#bdb1eed686184a9a4099efa772be7bc7"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_comRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Exports.comRingType"><span class="id" title="abbreviation">comRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#57b384122345a94c564987d4b6ee9f0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#57b384122345a94c564987d4b6ee9f0f"><span class="id" title="notation">comRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#57b384122345a94c564987d4b6ee9f0f"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#57b384122345a94c564987d4b6ee9f0f"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_comUnitRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Exports.comUnitRingType"><span class="id" title="abbreviation">comUnitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">comUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#e3ee791c903b0283e51d52d0692558ec"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_idomainType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Exports.idomainType"><span class="id" title="abbreviation">idomainType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#9894f8fff6e44a40eb9fd9cfcbde7780"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9894f8fff6e44a40eb9fd9cfcbde7780"><span class="id" title="notation">idomainType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#9894f8fff6e44a40eb9fd9cfcbde7780"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#9894f8fff6e44a40eb9fd9cfcbde7780"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_fieldType</span> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) <span class="id" title="var">charFp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.ssralg.html#005edfce3bb0bbe988e3333ca30adc0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#005edfce3bb0bbe988e3333ca30adc0f"><span class="id" title="notation">fieldType</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#005edfce3bb0bbe988e3333ca30adc0f"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.field.finfield.html#charFp"><span class="id" title="variable">charFp</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#005edfce3bb0bbe988e3333ca30adc0f"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="PrimeChar.FinRing"><span class="id" title="section">FinRing</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="PrimeChar.FinRing.R0"><span class="id" title="variable">R0</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Ring.Exports.finRingType"><span class="id" title="abbreviation">finRingType</span></a>) (<a name="PrimeChar.FinRing.charRp"><span class="id" title="variable">charRp</span></a> : <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#R0"><span class="id" title="variable">R0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finType</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#eb1f7a25d8e03f1f02a5769831d0e74e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#eb1f7a25d8e03f1f02a5769831d0e74e"><span class="id" title="notation">finType</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#eb1f7a25d8e03f1f02a5769831d0e74e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#eb1f7a25d8e03f1f02a5769831d0e74e"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finZmodType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">finZmodType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#2980bb304205aec85bc1eeb5d0a573a5"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_baseGroupType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">baseFinGroupType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ee332ddd6e3626489ee70ea4c624f1cd"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_groupType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">finGroupType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">+%</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#ad4d9ed93eeed8e8e57c81c6e35699c4"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finRingType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">finRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#cf58bd711195f609ec57107fc402496c"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finLmodType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#73928e07fdf596d7d0d44cccaf9a9cb3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#73928e07fdf596d7d0d44cccaf9a9cb3"><span class="id" title="notation">finLmodType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#73928e07fdf596d7d0d44cccaf9a9cb3"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#73928e07fdf596d7d0d44cccaf9a9cb3"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finLalgType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#f791552e58608a16cb248da4e7f34691"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f791552e58608a16cb248da4e7f34691"><span class="id" title="notation">finLalgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#f791552e58608a16cb248da4e7f34691"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f791552e58608a16cb248da4e7f34691"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finAlgType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#1f3938acab41e5853e751c34c441bf83"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#1f3938acab41e5853e751c34c441bf83"><span class="id" title="notation">finAlgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#1f3938acab41e5853e751c34c441bf83"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.algebra.finalg.html#1f3938acab41e5853e751c34c441bf83"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Let</span> <a name="PrimeChar.FinRing.pr_p"><span class="id" title="variable">pr_p</span></a> : <a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a>.  <br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_abelem"><span class="id" title="lemma">primeChar_abelem</span></a> : <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.abelian.html#bcb4124a3d9b102768b81d5d3006e029"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.abelian.html#bcb4124a3d9b102768b81d5d3006e029"><span class="id" title="notation">abelem</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_pgroup"><span class="id" title="lemma">primeChar_pgroup</span></a> : <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#5b9c9ef075a2fca9df30ee4ac4a1af18"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.solvable.pgroup.html#5b9c9ef075a2fca9df30ee4ac4a1af18"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">set</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#26c09fa7b21f5311d68f07b2527cd1eb"><span class="id" title="notation">]</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="order_primeChar"><span class="id" title="lemma">order_primeChar</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="mathcomp.ssreflect.eqtype.html#9e45f909d1732d6d9e153b650829bccf"><span class="id" title="notation">:&gt;</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">]</span></a>%<span class="id" title="var">g</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Let</span> <a name="PrimeChar.FinRing.n"><span class="id" title="variable">n</span></a> := <a class="idref" href="mathcomp.ssreflect.prime.html#logn"><span class="id" title="definition">logn</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="card_primeChar"><span class="id" title="lemma">card_primeChar</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinRing.n"><span class="id" title="variable">n</span></a>)%<span class="id" title="var">N</span>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_vectAxiom"><span class="id" title="lemma">primeChar_vectAxiom</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.axiom"><span class="id" title="abbreviation">Vector.axiom</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinRing.n"><span class="id" title="variable">n</span></a> (<a class="idref" href="mathcomp.field.finfield.html#primeChar_lmodType"><span class="id" title="definition">primeChar_lmodType</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinRing.charRp"><span class="id" title="variable">charRp</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="primeChar_vectMixin"><span class="id" title="definition">primeChar_vectMixin</span></a> := <a class="idref" href="mathcomp.algebra.vector.html#Vector.Mixin"><span class="id" title="constructor">Vector.Mixin</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_vectAxiom"><span class="id" title="lemma">primeChar_vectAxiom</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_vectType</span> := <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.VectType"><span class="id" title="abbreviation">VectType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="abbreviation">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#primeChar_vectMixin"><span class="id" title="definition">primeChar_vectMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="primeChar_dimf"><span class="id" title="lemma">primeChar_dimf</span></a> : <a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">dim</span></a> <a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">{:</span></a><a class="idref" href="mathcomp.field.finfield.html#primeChar_vectType"><span class="id" title="definition">primeChar_vectType</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinRing.n"><span class="id" title="variable">n</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinRing"><span class="id" title="section">FinRing</span></a>.<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finUnitRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.UnitRing.Exports.finUnitRingType"><span class="id" title="abbreviation">finUnitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">finUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.finalg.html#7f21453830587186138043335ab91dd1"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finUnitAlgType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.UnitRing.Exports.finUnitRingType"><span class="id" title="abbreviation">finUnitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#9af0def31728327fea663946c68b8952"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#9af0def31728327fea663946c68b8952"><span class="id" title="notation">finUnitAlgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#9af0def31728327fea663946c68b8952"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.finalg.html#9af0def31728327fea663946c68b8952"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_FalgType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.UnitRing.Exports.finUnitRingType"><span class="id" title="abbreviation">finUnitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.field.falgebra.html#3c0387428f19a365dfa0c989db9030d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.field.falgebra.html#3c0387428f19a365dfa0c989db9030d7"><span class="id" title="notation">FalgType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.falgebra.html#3c0387428f19a365dfa0c989db9030d7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.field.falgebra.html#3c0387428f19a365dfa0c989db9030d7"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finComRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.ComRing.Exports.finComRingType"><span class="id" title="abbreviation">finComRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">finComRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.finalg.html#381777e14bce98b548cb274563c7fc56"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finComUnitRingType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.ComUnitRing.Exports.finComUnitRingType"><span class="id" title="abbreviation">finComUnitRingType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">finComUnitRingType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.finalg.html#f0aa4fcf143660f4378ecfead8f3fdda"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finIdomainType</span> (<span class="id" title="var">R</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.IntegralDomain.Exports.finIdomainType"><span class="id" title="abbreviation">finIdomainType</span></a>) <span class="id" title="var">charRp</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">finIdomainType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#type"><span class="id" title="abbreviation">type</span></a> <a class="idref" href="mathcomp.field.finfield.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a><a class="idref" href="mathcomp.algebra.finalg.html#6c49b73b4d6aa1a932fafe7684bba39c"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="PrimeChar.FinField"><span class="id" title="section">FinField</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="PrimeChar.FinField.F0"><span class="id" title="variable">F0</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>) (<a name="PrimeChar.FinField.charFp"><span class="id" title="variable">charFp</span></a> : <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#F0"><span class="id" title="variable">F0</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>).<br/>

<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_finFieldType</span> := <a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">finFieldType</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="abbreviation">F</span></a><a class="idref" href="mathcomp.algebra.finalg.html#07fdfbae2c02044f4dae6b5dbeb0c7c7"><span class="id" title="notation">]</span></a>.<br/>
</div>

<div class="doc">
 We need to use the eta-long version of the constructor here as projections 
 of the Canonical fieldType of F cannot be computed syntactically.           
</div>
<div class="code">
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_fieldExtType</span> := <a class="idref" href="mathcomp.field.fieldext.html#5f5a3c95feae5e889ef0ed2ea21bd611"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.field.fieldext.html#5f5a3c95feae5e889ef0ed2ea21bd611"><span class="id" title="notation">fieldExtType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.fieldext.html#5f5a3c95feae5e889ef0ed2ea21bd611"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="abbreviation">F</span></a> <a class="idref" href="mathcomp.field.fieldext.html#5f5a3c95feae5e889ef0ed2ea21bd611"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinField.F0"><span class="id" title="variable">F0</span></a><a class="idref" href="mathcomp.field.fieldext.html#5f5a3c95feae5e889ef0ed2ea21bd611"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">primeChar_splittingFieldType</span> := <a class="idref" href="mathcomp.field.finfield.html#FinSplittingFieldType"><span class="id" title="abbreviation">FinSplittingFieldType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="abbreviation">F</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar.FinField"><span class="id" title="section">FinField</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#PrimeChar"><span class="id" title="section">PrimeChar</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinSplittingField"><span class="id" title="section">FinSplittingField</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinSplittingField.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>.<br/>

<br/>
</div>

<div class="doc">
 By card_vspace order K = #|K| for any finType structure on L; however we   
 do not want to impose the FinVector instance here.                          
</div>
<div class="code">
<span class="id" title="keyword">Let</span> <a name="FinSplittingField.order"><span class="id" title="variable">order</span></a> (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.algebra.vector.html#Vector.Exports.vectType"><span class="id" title="abbreviation">vectType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.F"><span class="id" title="variable">F</span></a>) (<span class="id" title="var">K</span> : <a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">vspace</span></a> <a class="idref" href="mathcomp.field.finfield.html#L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.algebra.vector.html#ca0a177f6d6581a7f5199987cd7ee21c"><span class="id" title="notation">}</span></a>) := (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.vector.html#ee35a6780ccd60155a3be89dcb5fdb30"><span class="id" title="notation">dim</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a>)%<span class="id" title="var">N</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinSplittingField.FinGalois"><span class="id" title="section">FinGalois</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinSplittingField.FinGalois.L"><span class="id" title="variable">L</span></a> : <a class="idref" href="mathcomp.field.galois.html#SplittingField.Exports.splittingFieldType"><span class="id" title="abbreviation">splittingFieldType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.F"><span class="id" title="variable">F</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> (<span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.F"><span class="id" title="variable">F</span></a>) (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.FinGalois.L"><span class="id" title="variable">L</span></a>) (<span class="id" title="var">K</span> <span class="id" title="var">E</span> : <a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">subfield</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.FinGalois.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">}</span></a>).<br/>

<br/>
<span class="id" title="keyword">Let</span> <a name="FinSplittingField.FinGalois.galL"><span class="id" title="variable">galL</span></a> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.field.galois.html#galois"><span class="id" title="definition">galois</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">{:</span></a><a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.FinGalois.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fact</span> <a name="galLgen"><span class="id" title="lemma">galLgen</span></a> <span class="id" title="var">K</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">alpha</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#generator"><span class="id" title="definition">generator</span></a> <a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">Gal</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">{:</span></a><a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.FinGalois.L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.finfield.html#alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.field.finfield.html#alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.order"><span class="id" title="variable">order</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finField_galois"><span class="id" title="lemma">finField_galois</span></a> <span class="id" title="var">K</span> <span class="id" title="var">E</span> : (<a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.algebra.vector.html#755d11a7d5629bce3486e7cbadc915e7"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.field.finfield.html#E"><span class="id" title="variable">E</span></a>)%<span class="id" title="var">VS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.field.galois.html#galois"><span class="id" title="definition">galois</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.field.finfield.html#E"><span class="id" title="variable">E</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="finField_galois_generator"><span class="id" title="lemma">finField_galois_generator</span></a> <span class="id" title="var">K</span> <span class="id" title="var">E</span> :<br/>
&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.algebra.vector.html#755d11a7d5629bce3486e7cbadc915e7"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.field.finfield.html#E"><span class="id" title="variable">E</span></a>)%<span class="id" title="var">VS</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">alpha</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.solvable.cyclic.html#generator"><span class="id" title="definition">generator</span></a> <a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">Gal</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.finfield.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.field.galois.html#7f39fd713ca3f00fbfda8b71eae7e2e1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.finfield.html#alpha"><span class="id" title="variable">alpha</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.finfield.html#E"><span class="id" title="variable">E</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.field.finfield.html#alpha"><span class="id" title="variable">alpha</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.order"><span class="id" title="variable">order</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.FinGalois"><span class="id" title="section">FinGalois</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="Fermat's_little_theorem"><span class="id" title="lemma">Fermat's_little_theorem</span></a> (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.field.fieldext.html#FieldExt.Exports.fieldExtType"><span class="id" title="abbreviation">fieldExtType</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.F"><span class="id" title="variable">F</span></a>) (<span class="id" title="var">K</span> : <a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">subfield</span></a> <a class="idref" href="mathcomp.field.finfield.html#L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.field.fieldext.html#da0a594fae595c8172b1a3e2dd69d19d"><span class="id" title="notation">}</span></a>) <span class="id" title="var">a</span> :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField.order"><span class="id" title="variable">order</span></a> <a class="idref" href="mathcomp.field.finfield.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.field.finfield.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinSplittingField"><span class="id" title="section">FinSplittingField</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinFieldExists"><span class="id" title="section">FinFieldExists</span></a>.<br/>
</div>

<div class="doc">
 While he existence of finite splitting fields and of finite fields of      
 arbitrary prime power order is mathematically straightforward, it is       
 technically challenging to formalize in Coq. The Coq typechecker performs  
 poorly for the spme of the deeply nested dependent types used in the       
 construction, such as polynomials over extensions of extensions of finite  
 fields. Any conversion in a nested structure parameter incurs a huge       
 overhead as it is shared across term comparison by call-by-need evalution. 
 The proof of FinSplittingFieldFor is contrived to mitigate this effect:    
 the abbreviation map_poly_extField alone divides by 3 the proof checking   
 time, by reducing the number of occurrences of field(Ext)Type structures   
 in the subgoals; the succesive, apparently redundant 'suffices' localize   
 some of the conversions to smaller subgoals, yielding a further 8-fold     
 time gain. In particular, we construct the splitting field as a subtype    
 of a recursive construction rather than prove that the latter yields       
 precisely a splitting field.                                                
<div class="paragraph"> </div>

  The apparently redundant type annotation reduces checking time by 30%.      
</div>
<div class="code">
<span class="id" title="keyword">Let</span> <a name="FinFieldExists.map_poly_extField"><span class="id" title="variable">map_poly_extField</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<span class="id" title="var">L</span> : <a class="idref" href="mathcomp.field.fieldext.html#FieldExt.Exports.fieldExtType"><span class="id" title="abbreviation">fieldExtType</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="variable">F</span></a>) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> (<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Theory.in_alg"><span class="id" title="abbreviation">in_alg</span></a> <a class="idref" href="mathcomp.field.finfield.html#L"><span class="id" title="variable">L</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.field.finfield.html#L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="FinSplittingFieldFor"><span class="id" title="lemma">FinSplittingFieldFor</span></a> (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a>) (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">poly</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="variable">F</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">L</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.field.galois.html#SplittingField.Exports.splittingFieldType"><span class="id" title="abbreviation">splittingFieldType</span></a> <a class="idref" href="mathcomp.field.finfield.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.field.galois.html#splittingFieldFor"><span class="id" title="definition">splittingFieldFor</span></a> 1 <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="mathcomp.field.finfield.html#445fdb3ebc0ff682c0cd1af9d3a32b17"><span class="id" title="notation">^%:</span></a><a class="idref" href="mathcomp.field.finfield.html#445fdb3ebc0ff682c0cd1af9d3a32b17"><span class="id" title="notation">A</span></a> <a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">{:</span></a><a class="idref" href="mathcomp.field.finfield.html#L"><span class="id" title="variable">L</span></a><a class="idref" href="mathcomp.algebra.vector.html#899a5fd19c4f3564d9757a9ac446b1dc"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="PrimePowerField"><span class="id" title="lemma">PrimePowerField</span></a> <span class="id" title="var">p</span> <span class="id" title="var">k</span> (<span class="id" title="var">m</span> := (<a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.field.finfield.html#k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">N</span>) :<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.prime.html#prime"><span class="id" title="definition">prime</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 0 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.field.finfield.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">Fm</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#Fm"><span class="id" title="variable">Fm</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.field.finfield.html#Fm"><span class="id" title="variable">Fm</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.field.finfield.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinFieldExists"><span class="id" title="section">FinFieldExists</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="FinDomain"><span class="id" title="section">FinDomain</span></a>.<br/>

<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">ssrnum</span> <span class="id" title="var">ssrint</span> <span class="id" title="var">algC</span> <span class="id" title="var">cyclotomic</span> <span class="id" title="var">Num.Theory</span>.<br/>
<span class="comment">(*&nbsp;Hide&nbsp;polynomial&nbsp;divisibility.&nbsp;*)</span><br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="FinDomain.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.UnitRing.Exports.finUnitRingType"><span class="id" title="abbreviation">finUnitRingType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="FinDomain.domR"><span class="id" title="variable">domR</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.axiom"><span class="id" title="definition">GRing.IntegralDomain.axiom</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a>.<br/>

<br/>
<span class="id" title="keyword">Let</span> <a name="FinDomain.lregR"><span class="id" title="variable">lregR</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 0 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.lreg"><span class="id" title="definition">GRing.lreg</span></a> <a class="idref" href="mathcomp.field.finfield.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="finDomain_field"><span class="id" title="lemma">finDomain_field</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.mixin_of"><span class="id" title="definition">GRing.Field.mixin_of</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a>.<br/>

<br/>
</div>

<div class="doc">
 This is Witt's proof of Wedderburn's little theorem.  
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a name="finDomain_mulrC"><span class="id" title="lemma">finDomain_mulrC</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">*%</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#d5d4e2467843f67554f1a8a22d125de9"><span class="id" title="notation">R</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="FinDomainFieldType"><span class="id" title="definition">FinDomainFieldType</span></a> : <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Exports.finFieldType"><span class="id" title="abbreviation">finFieldType</span></a> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">fin_unit_class</span> := <a class="idref" href="mathcomp.algebra.finalg.html#FinRing.UnitRing.class"><span class="id" title="definition">FinRing.UnitRing.class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">com_class</span> := <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComRing.Class"><span class="id" title="constructor">GRing.ComRing.Class</span></a> <a class="idref" href="mathcomp.field.finfield.html#finDomain_mulrC"><span class="id" title="lemma">finDomain_mulrC</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">com_unit_class</span> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.ComUnitRing.Class"><span class="id" title="constructor">GRing.ComUnitRing.Class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#com_class"><span class="id" title="variable">com_class</span></a> <a class="idref" href="mathcomp.field.finfield.html#fin_unit_class"><span class="id" title="variable">fin_unit_class</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">dom_class</span> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.IntegralDomain.Class"><span class="id" title="constructor">GRing.IntegralDomain.Class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#com_unit_class"><span class="id" title="variable">com_unit_class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.domR"><span class="id" title="variable">domR</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">field_class</span> := @<a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Class"><span class="id" title="constructor">GRing.Field.Class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#dom_class"><span class="id" title="variable">dom_class</span></a> <a class="idref" href="mathcomp.field.finfield.html#finDomain_field"><span class="id" title="lemma">finDomain_field</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">finfield_class</span> := @<a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Class"><span class="id" title="constructor">FinRing.Field.Class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.finfield.html#field_class"><span class="id" title="variable">field_class</span></a> <a class="idref" href="mathcomp.field.finfield.html#fin_unit_class"><span class="id" title="variable">fin_unit_class</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.algebra.finalg.html#FinRing.Field.Pack"><span class="id" title="constructor">FinRing.Field.Pack</span></a> <a class="idref" href="mathcomp.field.finfield.html#finfield_class"><span class="id" title="variable">finfield_class</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="FinDomainSplittingFieldType"><span class="id" title="definition">FinDomainSplittingFieldType</span></a> <span class="id" title="var">p</span> (<span class="id" title="var">charRp</span> : <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">char</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#b8d1051ec5bf038cb2a33edc541359f8"><span class="id" title="notation">]</span></a>) :=<br/>
&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="var">RoverFp</span> := @<a class="idref" href="mathcomp.field.finfield.html#primeChar_splittingFieldType"><span class="id" title="definition">primeChar_splittingFieldType</span></a> <a class="idref" href="mathcomp.field.finfield.html#p"><span class="id" title="variable">p</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomainFieldType"><span class="id" title="definition">FinDomainFieldType</span></a> <a class="idref" href="mathcomp.field.finfield.html#charRp"><span class="id" title="variable">charRp</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.field.galois.html#201f8b6ebe31f6a88a3d073a45335fc2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.field.galois.html#201f8b6ebe31f6a88a3d073a45335fc2"><span class="id" title="notation">splittingFieldType</span></a> <a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.algebra.zmodp.html#ac70144de8117a1d767eef28420399d1"><span class="id" title="notation">F_p</span></a> <a class="idref" href="mathcomp.field.galois.html#201f8b6ebe31f6a88a3d073a45335fc2"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.finfield.html#FinDomain.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.field.galois.html#201f8b6ebe31f6a88a3d073a45335fc2"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.field.finfield.html#RoverFp"><span class="id" title="variable">RoverFp</span></a><a class="idref" href="mathcomp.field.galois.html#201f8b6ebe31f6a88a3d073a45335fc2"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.finfield.html#FinDomain"><span class="id" title="section">FinDomain</span></a>.<br/>
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