path: root/mathcomp/character/finfield.v

(* (c) Copyright 2006-2015 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
Require Import mathcomp.ssreflect.ssreflect.
From mathcomp
Require Import ssrfun ssrbool eqtype ssrnat seq choice fintype div.
From mathcomp
Require Import tuple bigop prime finset fingroup ssralg poly polydiv.
From mathcomp
Require Import morphism action finalg zmodp cyclic center pgroup abelian.
From mathcomp
Require Import matrix mxabelem vector falgebra fieldext separable galois.
From mathcomp
Require ssrnum ssrint algC cyclotomic.

(******************************************************************************)
(*  Additional constructions and results on finite fields.                    *)
(*                                                                            *)
(*         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.              *)
(*   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.  *)
(******************************************************************************)

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Import GroupScope GRing.Theory FinRing.Theory.
Local Open Scope ring_scope.

Section FinRing.

Variable R : finRingType.

(* GG: Coq v8.3 fails to unify FinGroup.arg_sort _ with FinRing.sort R here   *)
(* because it expands the latter rather than FinGroup.arg_sort, which would   *)
(* expose the FinGroup.sort projection, thereby enabling canonical structure  *)
(* expansion. We should check whether the improved heuristics in Coq 8.4 have *)
(* resolved this issue.                                                       *)
Lemma finRing_nontrivial : [set: R] != 1%g.
Proof. by apply/(@trivgPn R); exists 1; rewrite ?inE ?oner_neq0. Qed.

(* GG: same issue here. *)
Lemma finRing_gt1 : 1 < #|R|.
Proof. by rewrite -cardsT (@cardG_gt1 R) finRing_nontrivial. Qed.

End FinRing.

Section FinField.

Variable F : finFieldType.

Lemma card_finField_unit : #|[set: {unit F}]| = #|F|.-1.
Proof.
by rewrite -(cardC1 0) cardsT card_sub; apply: eq_card => x; rewrite unitfE.
Qed.

Fact finField_unit_subproof x : x != 0 :> F -> x \is a GRing.unit.
Proof. by rewrite unitfE. Qed.

Definition finField_unit x nz_x :=
  FinRing.unit F (@finField_unit_subproof x nz_x).

Lemma expf_card x : x ^+ #|F| = x :> F.
Proof.
apply/eqP; rewrite -{2}[x]mulr1 -[#|F|]prednK; last by rewrite (cardD1 x).
rewrite exprS -subr_eq0 -mulrBr mulf_eq0 -implyNb -unitfE subr_eq0.
apply/implyP=> Ux; rewrite -(val_unitX _ (FinRing.unit F Ux)) -val_unit1.
by rewrite val_eqE -order_dvdn -card_finField_unit order_dvdG ?inE.
Qed.

Lemma finField_genPoly : 'X^#|F| - 'X = \prod_x ('X - x%:P) :> {poly F}.
Proof.
set n := #|F|; set Xn := 'X^n; set NX := - 'X; set pF := Xn + NX.
have lnNXn: size NX <= n by rewrite size_opp size_polyX finRing_gt1.
have UeF: uniq_roots (enum F) by rewrite uniq_rootsE enum_uniq.
rewrite [pF](all_roots_prod_XsubC _ _ UeF) ?size_addl ?size_polyXn -?cardE //.
  by rewrite enumT lead_coefDl ?size_polyXn // (monicP (monicXn _ _)) scale1r.
by apply/allP=> x _; rewrite rootE !hornerE hornerXn expf_card subrr.
Qed.

Lemma finCharP : {p | prime p & p \in [char F]}.
Proof.
pose e := exponent [set: F]; have e_gt0: e > 0 := exponent_gt0 _.
have: e%:R == 0 :> F by rewrite -zmodXgE expg_exponent // inE.
by case/natf0_char/sigW=> // p charFp; exists p; rewrite ?(charf_prime charFp).
Qed.

Lemma finField_is_abelem : is_abelem [set: F].
Proof.
have [p pr_p charFp] := finCharP.
apply/is_abelemP; exists p; rewrite ?abelemE ?zmod_abelian //=.
by apply/exponentP=> x _; rewrite zmodXgE mulrn_char.
Qed.

Lemma card_finCharP p n : #|F| = (p ^ n)%N -> prime p -> p \in [char F].
Proof.
move=> oF pr_p; rewrite inE pr_p -order_dvdn.
rewrite (abelem_order_p finField_is_abelem) ?inE ?oner_neq0 //=.
have n_gt0: n > 0 by rewrite -(ltn_exp2l _ _ (prime_gt1 pr_p)) -oF finRing_gt1.
by rewrite cardsT oF -(prednK n_gt0) pdiv_pfactor.
Qed. 

End FinField.

Section CardVspace.

Variables (F : finFieldType) (T : finType).

Section Vector.

Variable cvT : Vector.class_of F T.
Let vT := Vector.Pack (Phant F) cvT T.

Lemma card_vspace (V : {vspace vT}) : #|V| = (#|F| ^ \dim V)%N.
Proof.
set n := \dim V; pose V2rV v := \row_i coord (vbasis V) i v.
pose rV2V (rv : 'rV_n) := \sum_i rv 0 i *: (vbasis V)`_i.
have rV2V_K: cancel rV2V V2rV.
  have freeV: free (vbasis V) := basis_free (vbasisP V).
  by move=> rv; apply/rowP=> i; rewrite mxE coord_sum_free.
rewrite -[n]mul1n -card_matrix -(card_imset _ (can_inj rV2V_K)).
apply: eq_card => v; apply/idP/imsetP=> [/coord_vbasis-> | [rv _ ->]].
  by exists (V2rV v) => //; apply: eq_bigr => i _; rewrite mxE.
by apply: (@rpred_sum vT) => i _; rewrite rpredZ ?vbasis_mem ?memt_nth.
Qed.

Lemma card_vspacef : #|{: vT}%VS| = #|T|.
Proof. by apply: eq_card => v; rewrite (@memvf _ vT). Qed.

End Vector.

Variable caT : Falgebra.class_of F T.
Let aT := Falgebra.Pack (Phant F) caT T.

Lemma card_vspace1 : #|(1%VS : {vspace aT})| = #|F|.
Proof. by rewrite card_vspace (dimv1 aT). Qed.

End CardVspace.

Lemma VectFinMixin (R : finRingType) (vT : vectType R) : Finite.mixin_of vT.
Proof.
have v2rK := @Vector.InternalTheory.v2rK R vT.
exact: CanFinMixin (v2rK : @cancel _ (CountType vT (CanCountMixin v2rK)) _ _).
Qed.

(* These instacnces are not exported by default because they conflict with    *)
(* existing finType instances such as matrix_finType or primeChar_finType.    *)
Module FinVector.
Section Interfaces.

Variable F : finFieldType.
Implicit Types (vT : vectType F) (aT : FalgType F) (fT : fieldExtType F).

Canonical vect_finType vT := FinType vT (VectFinMixin vT).
Canonical Falg_finType aT := FinType aT (VectFinMixin aT).
Canonical fieldExt_finType fT := FinType fT (VectFinMixin fT).

Canonical Falg_finRingType aT := [finRingType of aT].
Canonical fieldExt_finRingType fT := [finRingType of fT].
Canonical fieldExt_finFieldType fT := [finFieldType of fT].

Lemma finField_splittingField_axiom fT : SplittingField.axiom fT.
Proof.
exists ('X^#|fT| - 'X); first by rewrite rpredB 1?rpredX ?polyOverX.
exists (enum fT); first by rewrite enumT finField_genPoly eqpxx.
by apply/vspaceP=> x; rewrite memvf seqv_sub_adjoin ?mem_enum.
Qed.

End Interfaces.
End FinVector.

Notation FinFieldExtType := FinVector.fieldExt_finFieldType.
Notation FinSplittingFieldAxiom := (FinVector.finField_splittingField_axiom _).
Notation FinSplittingFieldType F L :=
  (SplittingFieldType F L FinSplittingFieldAxiom).

Section PrimeChar.

Variable p : nat.

Section PrimeCharRing.

Variable R0 : ringType.

Definition PrimeCharType of p \in [char R0] : predArgType := R0.

Hypothesis charRp : p \in [char R0].
Local Notation R := (PrimeCharType charRp).
Implicit Types (a b : 'F_p) (x y : R).

Canonical primeChar_eqType := [eqType of R].
Canonical primeChar_choiceType := [choiceType of R].
Canonical primeChar_zmodType := [zmodType of R].
Canonical primeChar_ringType := [ringType of R].

Definition primeChar_scale a x := a%:R * x.
Local Infix "*p:" := primeChar_scale (at level 40).

Let natrFp n : (inZp n : 'F_p)%:R = n%:R :> R.
Proof.
rewrite {2}(divn_eq n p) natrD mulrnA (mulrn_char charRp) add0r.
by rewrite /= (Fp_cast (charf_prime charRp)).
Qed.

Lemma primeChar_scaleA a b x : a *p: (b *p: x) = (a * b) *p: x.
Proof. by rewrite /primeChar_scale mulrA -natrM natrFp. Qed.

Lemma primeChar_scale1 : left_id 1 primeChar_scale.
Proof. by move=> x; rewrite /primeChar_scale mul1r. Qed.

Lemma primeChar_scaleDr : right_distributive primeChar_scale +%R.
Proof. by move=> a x y /=; rewrite /primeChar_scale mulrDr. Qed.

Lemma primeChar_scaleDl x : {morph primeChar_scale^~ x: a b / a + b}.
Proof. by move=> a b; rewrite /primeChar_scale natrFp natrD mulrDl. Qed.

Definition primeChar_lmodMixin :=
  LmodMixin primeChar_scaleA primeChar_scale1
            primeChar_scaleDr primeChar_scaleDl.
Canonical primeChar_lmodType := LmodType 'F_p R primeChar_lmodMixin.

Lemma primeChar_scaleAl : GRing.Lalgebra.axiom ( *%R : R -> R -> R).
Proof. by move=> a x y; apply: mulrA. Qed.
Canonical primeChar_LalgType := LalgType 'F_p R primeChar_scaleAl.

Lemma primeChar_scaleAr : GRing.Algebra.axiom primeChar_LalgType.
Proof. by move=> a x y; rewrite ![a *: _]mulr_natl mulrnAr. Qed.
Canonical primeChar_algType := AlgType 'F_p R primeChar_scaleAr.

End PrimeCharRing.

Local Notation type := @PrimeCharType.

Canonical primeChar_unitRingType (R : unitRingType) charRp :=
  [unitRingType of type R charRp].
Canonical primeChar_unitAlgType (R : unitRingType) charRp :=
  [unitAlgType 'F_p of type R charRp].
Canonical primeChar_comRingType (R : comRingType) charRp :=
  [comRingType of type R charRp].
Canonical primeChar_comUnitRingType (R : comUnitRingType) charRp :=
  [comUnitRingType of type R charRp].
Canonical primeChar_idomainType (R : idomainType) charRp :=
  [idomainType of type R charRp].
Canonical primeChar_fieldType (F : fieldType) charFp :=
  [fieldType of type F charFp].

Section FinRing.

Variables (R0 : finRingType) (charRp : p \in [char R0]).
Local Notation R := (type _ charRp).

Canonical primeChar_finType := [finType of R].
Canonical primeChar_finZmodType := [finZmodType of R].
Canonical primeChar_baseGroupType := [baseFinGroupType of R for +%R].
Canonical primeChar_groupType := [finGroupType of R for +%R].
Canonical primeChar_finRingType := [finRingType of R].

Let pr_p : prime p. Proof. exact: charf_prime charRp. Qed.

Lemma primeChar_abelem : p.-abelem [set: R].
Proof.
rewrite abelemE ?zmod_abelian //=.
by apply/exponentP=> x _; rewrite zmodXgE mulrn_char.
Qed.

Lemma primeChar_pgroup : p.-group [set: R].
Proof. by case/and3P: primeChar_abelem. Qed.

Lemma order_primeChar x : x != 0 :> R -> #[x]%g = p.
Proof. by apply: (abelem_order_p primeChar_abelem); rewrite inE. Qed.

Let n := logn p #|R|.

Lemma card_primeChar : #|R| = (p ^ n)%N.
Proof. by rewrite /n -cardsT {1}(card_pgroup primeChar_pgroup). Qed.

Lemma primeChar_vectAxiom : Vector.axiom n (primeChar_lmodType charRp).
Proof.
have /isog_isom/=[f /isomP[injf im_f]]: [set: R] \isog [set: 'rV['F_p]_n].
  have [abelR ntR] := (primeChar_abelem, finRing_nontrivial R0).
  by rewrite /n -cardsT -(dim_abelemE abelR) ?isog_abelem_rV.
exists f; last by exists (invm injf) => x; rewrite ?invmE ?invmK ?im_f ?inE.
move=> a x y; rewrite [a *: _]mulr_natl morphM ?morphX ?inE // zmodXgE.
by congr (_ + _); rewrite -scaler_nat natr_Zp.
Qed.

Definition primeChar_vectMixin := Vector.Mixin primeChar_vectAxiom.
Canonical primeChar_vectType := VectType 'F_p R primeChar_vectMixin.

Lemma primeChar_dimf : \dim {:primeChar_vectType} = n.
Proof. by rewrite dimvf. Qed.

End FinRing.

Canonical primeChar_finUnitRingType (R : finUnitRingType) charRp :=
  [finUnitRingType of type R charRp].
Canonical primeChar_finUnitAlgType (R : finUnitRingType) charRp :=
  [finUnitAlgType 'F_p of type R charRp].
Canonical primeChar_FalgType (R : finUnitRingType) charRp :=
  [FalgType 'F_p of type R charRp].
Canonical primeChar_finComRingType (R : finComRingType) charRp :=
  [finComRingType of type R charRp].
Canonical primeChar_finComUnitRingType (R : finComUnitRingType) charRp :=
  [finComUnitRingType of type R charRp].
Canonical primeChar_finIdomainType (R : finIdomainType) charRp :=
  [finIdomainType of type R charRp].

Section FinField.

Variables (F0 : finFieldType) (charFp : p \in [char F0]).
Local Notation F := (type _ charFp).

Canonical primeChar_finFieldType := [finFieldType of F].
(* We need to use the eta-long version of the constructor here as projections *)
(* of the Canonical fieldType of F cannot be computed syntactically.          *)
Canonical primeChar_fieldExtType := [fieldExtType 'F_p of F for F0].
Canonical primeChar_splittingFieldType := FinSplittingFieldType 'F_p F.

End FinField.

End PrimeChar.

Section FinSplittingField.

Variable F : finFieldType.

(* By card_vspace order K = #|K| for any finType structure on L; however we   *)
(* do not want to impose the FinVector instance here.                         *)
Let order (L : vectType F) (K : {vspace L}) := (#|F| ^ \dim K)%N.

Section FinGalois.

Variable L : splittingFieldType F.
Implicit Types (a b : F) (x y : L) (K E : {subfield L}).

Let galL K : galois K {:L}.
Proof.
without loss {K} ->: K / K = 1%AS.
  by move=> IH; apply: galoisS (IH _ (erefl _)); rewrite sub1v subvf.
apply/splitting_galoisField; pose finL := FinFieldExtType L.
exists ('X^#|finL| - 'X); split; first by rewrite rpredB 1?rpredX ?polyOverX.
  rewrite (finField_genPoly finL) -big_filter.
  by rewrite separable_prod_XsubC ?(enum_uniq finL).
exists (enum finL); first by rewrite enumT (finField_genPoly finL) eqpxx.
by apply/vspaceP=> x; rewrite memvf seqv_sub_adjoin ?(mem_enum finL).
Qed.

Fact galLgen K :
  {alpha | generator 'Gal({:L} / K) alpha & forall x, alpha x = x ^+ order K}.
Proof.
without loss{K} ->: K / K = 1%AS; last rewrite /order dimv1 expn1.
  rewrite /generator => /(_ _ (erefl _))[alpha /eqP defGalL].
  rewrite /order dimv1 expn1 => Dalpha.
  exists (alpha ^+ \dim K)%g => [|x]; last first.
    elim: (\dim K) => [|n IHn]; first by rewrite gal_id.
    by rewrite expgSr galM ?memvf // IHn Dalpha expnSr exprM.
  rewrite (eq_subG_cyclic (cycle_cyclic alpha)) ?cycleX //=; last first.
    by rewrite -defGalL galS ?sub1v.
  rewrite eq_sym -orderE orderXdiv orderE -defGalL -{1}(galois_dim (galL 1)).
    by rewrite dimv1 divn1 galois_dim.
  by rewrite dimv1 divn1 field_dimS // subvf.
pose f x := x ^+ #|F|.
have idfP x: reflect (f x = x) (x \in 1%VS).
  rewrite /f; apply: (iffP (vlineP _ _)) => [[a ->] | xFx].
    by rewrite -in_algE -rmorphX expf_card.
  pose q := map_poly (in_alg L) ('X^#|F| - 'X).
  have: root q x.
    rewrite /q rmorphB /= map_polyXn map_polyX.
    by rewrite rootE !(hornerE, hornerXn) xFx subrr.
  have{q} ->: q = \prod_(z <- [seq b%:A | b : F]) ('X - z%:P).
    rewrite /q finField_genPoly rmorph_prod big_map enumT.
    by apply: eq_bigr => b _; rewrite rmorphB /= map_polyX map_polyC.
  by rewrite root_prod_XsubC => /mapP[a]; exists a.
have fM: rmorphism f.
  rewrite /f; do 2?split=> [x y|]; rewrite ?exprMn ?expr1n //.
  have [p _ charFp] := finCharP F; rewrite (card_primeChar charFp).
  elim: (logn _ _) => // n IHn; rewrite expnSr !exprM {}IHn.
  by rewrite -(char_lalg L) in charFp; rewrite -Frobenius_autE rmorphB.
have fZ: linear f.
  move=> a x y; rewrite -mulr_algl [f _](rmorphD (RMorphism fM)) rmorphM /=.
  by rewrite (idfP _ _) ?mulr_algl ?memvZ // memv_line.
have /kAut_to_gal[alpha galLalpha Dalpha]: kAut 1 {:L} (linfun (Linear fZ)).
  rewrite kAutfE; apply/kHomP; split=> [x y _ _ | x /idfP]; rewrite !lfunE //=.
  exact: (rmorphM (RMorphism fM)).
exists alpha => [|a]; last by rewrite -Dalpha ?memvf ?lfunE.
suffices <-: fixedField [set alpha] = 1%AS by rewrite gal_generated /generator.
apply/vspaceP => x; apply/fixedFieldP/idfP; rewrite ?memvf // => id_x.
  by rewrite -{2}(id_x _ (set11 _)) -Dalpha ?lfunE ?memvf.
by move=> _ /set1P->; rewrite -Dalpha ?memvf ?lfunE.
Qed.

Lemma finField_galois K E : (K <= E)%VS -> galois K E.
Proof.
move=> sKE; have /galois_fixedField <- := galL E.
rewrite normal_fixedField_galois // -sub_abelian_normal ?galS //.
apply: abelianS (galS _ (sub1v _)) _.
by have [alpha /('Gal(_ / _) =P _)-> _] := galLgen 1; apply: cycle_abelian.
Qed.

Lemma finField_galois_generator K E :
   (K <= E)%VS ->
 {alpha | generator 'Gal(E / K) alpha
        & {in E, forall x, alpha x = x ^+ order K}}.
Proof.
move=> sKE; have [alpha defGalLK Dalpha] := galLgen K.
have inKL_E: (K <= E <= {:L})%VS by rewrite sKE subvf.
have nKE: normalField K E by have/and3P[] := finField_galois sKE.
have galLKalpha: alpha \in 'Gal({:L} / K).
  by rewrite (('Gal(_ / _) =P _) defGalLK) cycle_id.
exists (normalField_cast _ alpha) => [|x Ex]; last first.
  by rewrite (normalField_cast_eq inKL_E).
rewrite /generator -(morphim_cycle (normalField_cast_morphism inKL_E nKE)) //.
by rewrite -((_ =P <[alpha]>) defGalLK) normalField_img.
Qed.

End FinGalois.

Lemma Fermat's_little_theorem (L : fieldExtType F) (K : {subfield L}) a :
  (a \in K) = (a ^+ order K == a).
Proof.
move: K a; wlog [{L}L -> K a]: L / exists galL : splittingFieldType F, L = galL.
  by pose galL := (FinSplittingFieldType F L) => /(_ galL); apply; exists galL.
have /galois_fixedField fixLK := finField_galois (subvf K).
have [alpha defGalLK Dalpha] := finField_galois_generator (subvf K).
rewrite -Dalpha ?memvf // -{1}fixLK (('Gal(_ / _) =P _) defGalLK).
rewrite /cycle -gal_generated (galois_fixedField _) ?fixedField_galois //.
by apply/fixedFieldP/eqP=> [|-> | alpha_x _ /set1P->]; rewrite ?memvf ?set11.
Qed.

End FinSplittingField.

Section FinDomain.

Import ssrnum ssrint algC cyclotomic Num.Theory.
Local Infix "%|" := dvdn. (* Hide polynomial divisibility. *)

Variable R : finUnitRingType.

Hypothesis domR : GRing.IntegralDomain.axiom R.
Implicit Types x y : R.

Let lregR x : x != 0 -> GRing.lreg x.  
Proof. by move=> xnz; apply: mulrI0_lreg => y /domR/orP[/idPn | /eqP]. Qed.

Lemma finDomain_field : GRing.Field.mixin_of R. 
Proof.
move=> x /lregR-regx; apply/unitrP; exists (invF regx 1).
by split; first apply: (regx); rewrite ?mulrA f_invF // mulr1 mul1r.
Qed.

(* This is Witt's proof of Wedderburn's little theorem. *)
Theorem finDomain_mulrC : @commutative R R *%R.
Proof.
have fieldR := finDomain_field.
have [p p_pr charRp]: exists2 p, prime p & p \in [char R].
  have [e /prod_prime_decomp->]: {e | (e > 0)%N & e%:R == 0 :> R}.
    by exists #|[set: R]%G|; rewrite // -order_dvdn order_dvdG ?inE. 
  rewrite big_seq; elim/big_rec: _ => [|[p m] /= n]; first by rewrite oner_eq0.
  case/mem_prime_decomp=> p_pr _ _ IHn.
  elim: m => [|m IHm]; rewrite ?mul1n {IHn}// expnS -mulnA natrM.
  by case/eqP/domR/orP=> //; exists p; last apply/andP.
pose Rp := PrimeCharType charRp; pose L : {vspace Rp} := fullv.
pose G := [set: {unit R}]; pose ofG : {unit R} -> Rp := val.
pose projG (E : {vspace Rp}) := [preim ofG of E].
have inG t nzt: Sub t (finDomain_field nzt) \in G by rewrite inE.
have card_projG E: #|projG E| = (p ^ \dim E - 1)%N.
  transitivity #|E|.-1; last by rewrite subn1 card_vspace card_Fp.
  rewrite (cardD1 0) mem0v (card_preim val_inj) /=.
  apply: eq_card => x; congr (_ && _); rewrite [LHS]codom_val.
  by apply/idP/idP=> [/(memPn _ _)-> | /fieldR]; rewrite ?unitr0.
pose C u := 'C[ofG u]%AS; pose Q := 'C(L)%AS; pose q := (p ^ \dim Q)%N.
have defC u: 'C[u] =i projG (C u).
  by move=> v; rewrite cent1E !inE (sameP cent1vP eqP).
have defQ: 'Z(G) =i projG Q.
  move=> u; rewrite !inE.
  apply/centP/centvP=> cGu v _; last exact/val_inj/cGu/memvf.
  by have [-> | /inG/cGu[]] := eqVneq v 0; first by rewrite commr0.
have q_gt1: (1 < q)%N by rewrite (ltn_exp2l 0) ?prime_gt1 ?adim_gt0.
pose n := \dim_Q L; have oG: #|G| = (q ^ n - 1)%N.
  rewrite -expnM mulnC divnK ?skew_field_dimS ?subvf // -card_projG.
  by apply: eq_card => u; rewrite !inE memvf.
have oZ: #|'Z(G)| = (q - 1)%N by rewrite -card_projG; apply: eq_card.
suffices n_le1: (n <= 1)%N.
  move=> u v; apply/centvsP: (memvf (u : Rp)) (memvf (v : Rp)) => {u v}.
  rewrite -(geq_leqif (dimv_leqif_sup (subvf Q))) -/L.
  by rewrite leq_divLR ?mul1n ?skew_field_dimS ?subvf in n_le1.
without loss n_gt1: / (1 < n)%N by rewrite ltnNge; apply: wlog_neg.
have [q_gt0 n_gt0] := (ltnW q_gt1, ltnW n_gt1).
have [z z_prim] := C_prim_root_exists n_gt0.
have zn1: z ^+ n = 1 by apply: prim_expr_order.
have /eqP-n1z: `|z| == 1.
  by rewrite -(pexpr_eq1 n_gt0) ?normr_ge0 // -normrX zn1 normr1.
suffices /eqP/normC_sub_eq[t n1t [Dq Dz]]: `|q%:R - z| == `|q%:R| - `|z|.
  suffices z1: z == 1 by rewrite leq_eqVlt -dvdn1 (prim_order_dvd z_prim) z1.
  by rewrite Dz n1z mul1r -(eqr_pmuln2r q_gt0) Dq normr_nat mulr_natl.
pose aq d : algC := (cyclotomic (z ^+ (n %/ d)) d).[q%:R].
suffices: `|aq n| <= (q - 1)%:R.
  rewrite eqr_le ler_sub_dist andbT n1z normr_nat natrB //; apply: ler_trans.
  rewrite {}/aq horner_prod divnn n_gt0 expr1 normr_prod.
  rewrite (bigD1 (Ordinal n_gt1)) ?coprime1n //= !hornerE ler_pemulr //.
  elim/big_ind: _ => // [|d _]; first exact: mulr_ege1.
  rewrite !hornerE; apply: ler_trans (ler_sub_dist _ _).
  by rewrite normr_nat normrX n1z expr1n ler_subr_addl (leC_nat 2).
have Zaq d: d %| n -> aq d \in Cint.
  move/(dvdn_prim_root z_prim)=> zd_prim. 
  rewrite rpred_horner ?rpred_nat //= -Cintr_Cyclotomic //.
  by apply/polyOverP=> i; rewrite coef_map ?rpred_int.
suffices: (aq n %| (q - 1)%:R)%C.
  rewrite {1}[aq n]CintEsign ?Zaq // -(rpredMsign _ (aq n < 0)%R).
  rewrite dvdC_mul2l ?signr_eq0 //.
  have /CnatP[m ->]: `|aq n| \in Cnat by rewrite Cnat_norm_Cint ?Zaq.
  by rewrite leC_nat dvdC_nat; apply: dvdn_leq; rewrite subn_gt0.
have prod_aq m: m %| n -> \prod_(d < n.+1 | d %| m) aq d = (q ^ m - 1)%:R.
  move=> m_dv_n; transitivity ('X^m - 1).[q%:R : algC]; last first.
    by rewrite !hornerE hornerXn -natrX natrB ?expn_gt0 ?prime_gt0.
  rewrite (prod_cyclotomic (dvdn_prim_root z_prim m_dv_n)).
  have def_divm: perm_eq (divisors m) [seq d <- index_iota 0 n.+1 | d %| m].
    rewrite uniq_perm_eq ?divisors_uniq ?filter_uniq ?iota_uniq // => d.
    rewrite -dvdn_divisors ?(dvdn_gt0 n_gt0) // mem_filter mem_iota ltnS /=.
    by apply/esym/andb_idr=> d_dv_m; rewrite dvdn_leq ?(dvdn_trans d_dv_m).
  rewrite (eq_big_perm _ def_divm) big_filter big_mkord horner_prod.
  by apply: eq_bigr => d d_dv_m; rewrite -exprM muln_divA ?divnK.
have /rpredBl<-: (aq n %| #|G|%:R)%C.
  rewrite oG -prod_aq // (bigD1 ord_max) //= dvdC_mulr //.
  by apply: rpred_prod => d /andP[/Zaq].
rewrite center_class_formula addrC oZ natrD addKr natr_sum /=.
apply: rpred_sum => _ /imsetP[u /setDP[_ Z'u] ->]; rewrite -/G /=.
have sQC: (Q <= C u)%VS by apply/subvP=> v /centvP-cLv; apply/cent1vP/cLv/memvf.
have{sQC} /dvdnP[m Dm]: \dim Q %| \dim (C u) by apply: skew_field_dimS.
have m_dv_n: m %| n by rewrite dvdn_divRL // -?Dm ?skew_field_dimS ?subvf.
have m_gt0: (0 < m)%N := dvdn_gt0 n_gt0 m_dv_n.
have{Dm} oCu: #|'C[u]| = (q ^ m - 1)%N.
  by rewrite -expnM mulnC -Dm (eq_card (defC u)) card_projG.
have ->: #|u ^: G|%:R = \prod_(d < n.+1 | d %| n) (aq d / aq d ^+ (d %| m)).
  rewrite -index_cent1 natf_indexg ?subsetT //= setTI prodf_div prod_aq // -oG.
  congr (_ / _); rewrite big_mkcond oCu -prod_aq //= big_mkcond /=.
  by apply: eq_bigr => d _; case: ifP => [/dvdn_trans->| _]; rewrite ?if_same.
rewrite (bigD1 ord_max) //= [n %| m](contraNF _ Z'u) => [|n_dv_m]; last first.
  rewrite -sub_cent1 subEproper eq_sym eqEcard subsetT oG oCu leq_sub2r //.
  by rewrite leq_exp2l // dvdn_leq.
rewrite divr1 dvdC_mulr //; apply/rpred_prod => d /andP[/Zaq-Zaqd _].
have [-> | nz_aqd] := eqVneq (aq d) 0; first by rewrite mul0r.
by rewrite -[aq d]expr1 -exprB ?leq_b1 ?unitfE ?rpredX.
Qed.

Definition FinDomainFieldType : finFieldType :=
  let fin_unit_class := FinRing.UnitRing.class R in
  let com_class := GRing.ComRing.Class finDomain_mulrC in
  let com_unit_class := @GRing.ComUnitRing.Class R com_class fin_unit_class in
  let dom_class := @GRing.IntegralDomain.Class R com_unit_class domR in
  let field_class := @GRing.Field.Class R dom_class finDomain_field in
  let finfield_class := @FinRing.Field.Class R field_class fin_unit_class in
  FinRing.Field.Pack finfield_class R.

Definition FinDomainSplittingFieldType p (charRp : p \in [char R]) :=
   let RoverFp := @primeChar_splittingFieldType p FinDomainFieldType charRp in
   [splittingFieldType 'F_p of R for RoverFp].

End FinDomain.