blob: aa9112426d835bc08aa3047633e2ec033f464342 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
|
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>mathcomp.field.cyclotomic</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library mathcomp.field.cyclotomic</h1>
<div class="code">
<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/>
Distributed under the terms of CeCILL-B. *)</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">
This file provides few basic properties of cyclotomic polynomials.
We define:
cyclotomic z n == the factorization of the nth cyclotomic polynomial in
a ring R in which z is an nth primitive root of unity.
'Phi_n == the nth cyclotomic polynomial in int.
This library is quite limited, and should be extended in the future. In
particular the irreducibity of 'Phi_n is only stated indirectly, as the
fact that its embedding in the algebraics (algC) is the minimal polynomial
of an nth primitive root of unity.
</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">GRing.Theory</span> <span class="id" title="var">Num.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="CyclotomicPoly"><span class="id" title="section">CyclotomicPoly</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="CyclotomicPoly.Ring"><span class="id" title="section">Ring</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="CyclotomicPoly.Ring.R"><span class="id" title="variable">R</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="cyclotomic"><span class="id" title="definition">cyclotomic</span></a> (<span class="id" title="var">z</span> : <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Ring.R"><span class="id" title="variable">R</span></a>) <span class="id" title="var">n</span> :=<br/>
<a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">(</span></a><span class="id" title="var">k</span> <a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#k"><span class="id" title="variable">k</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#f2061c5b083fb574331c7bf65b44ceb4"><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.algebra.poly.html#5d46c3ff21505243f65fdae89313c246"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#k"><span class="id" title="variable">k</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#f2061c5b083fb574331c7bf65b44ceb4"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="cyclotomic_monic"><span class="id" title="lemma">cyclotomic_monic</span></a> <span class="id" title="var">z</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.field.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.field.cyclotomic.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.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_cyclotomic"><span class="id" title="lemma">size_cyclotomic</span></a> <span class="id" title="var">z</span> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> (<a class="idref" href="mathcomp.field.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</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.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#totient"><span class="id" title="definition">totient</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Ring"><span class="id" title="section">Ring</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="separable_Xn_sub_1"><span class="id" title="lemma">separable_Xn_sub_1</span></a> (<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">n</span> :<br/>
<a class="idref" href="mathcomp.field.cyclotomic.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="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">:></span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#R"><span class="id" title="variable">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.separable.html#separable_poly"><span class="id" title="definition">separable_poly</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#R"><span class="id" title="variable">R</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.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="CyclotomicPoly.Field"><span class="id" title="section">Field</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variables</span> (<a name="CyclotomicPoly.Field.F"><span class="id" title="variable">F</span></a> : <a class="idref" href="mathcomp.algebra.ssralg.html#GRing.Field.Exports.fieldType"><span class="id" title="abbreviation">fieldType</span></a>) (<a name="CyclotomicPoly.Field.n"><span class="id" title="variable">n</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>) (<a name="CyclotomicPoly.Field.z"><span class="id" title="variable">z</span></a> : <a class="idref" href="mathcomp.field.cyclotomic.html#F"><span class="id" title="variable">F</span></a>).<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="CyclotomicPoly.Field.prim_z"><span class="id" title="variable">prim_z</span></a> : <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.z"><span class="id" title="variable">z</span></a>.<br/>
<span class="id" title="keyword">Let</span> <a name="CyclotomicPoly.Field.n_gt0"><span class="id" title="variable">n_gt0</span></a> := <a class="idref" href="mathcomp.algebra.poly.html#prim_order_gt0"><span class="id" title="lemma">prim_order_gt0</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.prim_z"><span class="id" title="variable">prim_z</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="root_cyclotomic"><span class="id" title="lemma">root_cyclotomic</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.algebra.poly.html#root"><span class="id" title="definition">root</span></a> (<a class="idref" href="mathcomp.field.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a>) <a class="idref" href="mathcomp.field.cyclotomic.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.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="prod_cyclotomic"><span class="id" title="lemma">prod_cyclotomic</span></a> :<br/>
<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.field.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1 <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.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">d</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#divisors"><span class="id" title="definition">divisors</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> (<a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field.n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#df17451da28eb630dbb51b12706ba39e"><span class="id" title="notation">%/</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#d"><span class="id" title="variable">d</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#fb22424322c3d7eb9b837dfca65ce21e"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.field.cyclotomic.html#d"><span class="id" title="variable">d</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly.Field"><span class="id" title="section">Field</span></a>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.field.cyclotomic.html#CyclotomicPoly"><span class="id" title="section">CyclotomicPoly</span></a>.<br/>
<br/>
<br/>
<br/>
<span class="id" title="keyword">Local Hint Resolve</span> (@<a class="idref" href="mathcomp.algebra.ssrint.html#intr_inj"><span class="id" title="definition">intr_inj</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#faa7b03f15fa8c0b383b6f3802b37e9e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#faa7b03f15fa8c0b383b6f3802b37e9e"><span class="id" title="notation">numDomainType</span></a> <a class="idref" href="mathcomp.algebra.ssrnum.html#faa7b03f15fa8c0b383b6f3802b37e9e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</span></a><a class="idref" href="mathcomp.algebra.ssrnum.html#faa7b03f15fa8c0b383b6f3802b37e9e"><span class="id" title="notation">]</span></a>).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="C_prim_root_exists"><span class="id" title="lemma">C_prim_root_exists</span></a> <span class="id" title="var">n</span> : (<a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<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="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">z</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.algC.html#Algebraics.Exports.algC"><span class="id" title="abbreviation">algC</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.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</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/>
</div>
<div class="doc">
(Integral) Cyclotomic polynomials.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a name="Cyclotomic"><span class="id" title="definition">Cyclotomic</span></a> <span class="id" title="var">n</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.algebra.ssrint.html#int"><span class="id" title="inductive">int</span></a><a class="idref" href="mathcomp.algebra.poly.html#699040ddc0986f520cece215f531d947"><span class="id" title="notation">}</span></a> :=<br/>
<span class="id" title="keyword">let</span>: <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">z</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.field.cyclotomic.html#C_prim_root_exists"><span class="id" title="lemma">C_prim_root_exists</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#ltn0Sn"><span class="id" title="lemma">ltn0Sn</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a>) <span class="id" title="tactic">in</span><br/>
<a class="idref" href="mathcomp.algebra.poly.html#map_poly"><span class="id" title="definition">map_poly</span></a> <a class="idref" href="mathcomp.field.algC.html#Algebraics.Exports.floorC"><span class="id" title="definition">floorC</span></a> (<a class="idref" href="mathcomp.field.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <span class="id" title="var">z</span> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a>).<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">"</span></a>''Phi_' n" := (<a class="idref" href="mathcomp.field.cyclotomic.html#Cyclotomic"><span class="id" title="definition">Cyclotomic</span></a> <span class="id" title="var">n</span>)<br/>
(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">n</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">format</span> "''Phi_' n").<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="Cyclotomic_monic"><span class="id" title="lemma">Cyclotomic_monic</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">Phi_n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#e6408d45e92e642f7d1652448339ba09"><span class="id" title="notation">is</span></a> <a class="idref" href="mathcomp.algebra.poly.html#monic"><span class="id" title="definition">monic</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="Cintr_Cyclotomic"><span class="id" title="lemma">Cintr_Cyclotomic</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> :<br/>
<a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</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.cyclotomic.html#pZtoC"><span class="id" title="abbreviation">pZtoC</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">Phi_n</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.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="prod_Cyclotomic"><span class="id" title="lemma">prod_Cyclotomic</span></a> <span class="id" title="var">n</span> :<br/>
(<a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#19ab5cfd7e4f60fa14f22b576013bd96"><span class="id" title="notation">></span></a> 0)%<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.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">(</span></a><span class="id" title="var">d</span> <a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.ssreflect.prime.html#divisors"><span class="id" title="definition">divisors</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.ssralg.html#add995903469f3735748795c8f1b81bd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">Phi_d</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.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.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.algebra.ssralg.html#d70623330b2787db6b196e37db7d8f45"><span class="id" title="notation">-</span></a> 1.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="Cyclotomic0"><span class="id" title="lemma">Cyclotomic0</span></a> : <a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">Phi_0</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> 1.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="size_Cyclotomic"><span class="id" title="lemma">size_Cyclotomic</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#size"><span class="id" title="definition">size</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.field.cyclotomic.html#84e794f5513e5917c9e68cdf3e4a2b12"><span class="id" title="notation">Phi_n</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.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.prime.html#totient"><span class="id" title="definition">totient</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">).+1</span></a>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="minCpoly_cyclotomic"><span class="id" title="lemma">minCpoly_cyclotomic</span></a> <span class="id" title="var">n</span> <span class="id" title="var">z</span> :<br/>
<a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">.-</span></a><a class="idref" href="mathcomp.algebra.poly.html#92efb5ea268b6e2f9a125afe76aecbba"><span class="id" title="notation">primitive_root</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</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.algC.html#Algebraics.Exports.minCpoly"><span class="id" title="definition">minCpoly</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</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.cyclotomic.html#cyclotomic"><span class="id" title="definition">cyclotomic</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.field.cyclotomic.html#n"><span class="id" title="variable">n</span></a>.<br/>
</div>
</div>
<div id="footer">
<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a>
</div>
</div>
</body>
</html>
|
