fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
pprimeChar_scaleAr(a : 'F_p) (x y : R) : a *: (x * y) = x * (a *: y).
Proof. by rewrite ![a *: _]mulr_natl mulrnAr. Qed.
HB.instance Definition _ := GRing.Lalgebra_isAlgebra.Build 'F_p R
pprimeChar_scaleAr. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_scaleAr | |
Definition_ (R : unitRingType) pcharRp :=
GRing.UnitRing.on (type R pcharRp).
HB.instance Definition _ (R : comNzRingType) pcharRp :=
GRing.ComNzRing.on (type R pcharRp).
HB.instance Definition _ (R : comUnitRingType) pcharRp :=
GRing.ComUnitRing.on (type R pcharRp).
HB.instance Definition _ (R : idomainType) pcharRp :=
GRing.IntegralDomain.on (type R pcharRp).
HB.instance Definition _ (R : fieldType) pcharRp :=
GRing.Field.on (type R pcharRp). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
Definition_ := FinGroup.on R.
Let pr_p : prime p. Proof. exact: pcharf_prime pcharRp. Qed. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
pprimeChar_abelem: p.-abelem [set: R].
Proof. exact: fin_Fp_lmod_abelem. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_abelem | |
pprimeChar_pgroup: p.-group [set: R].
Proof. by case/and3P: pprimeChar_abelem. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_pgroup | |
order_pprimeCharx : x != 0 :> R -> #[x]%g = p.
Proof. by apply: (abelem_order_p pprimeChar_abelem); rewrite inE. Qed.
Let n := logn p #|R|. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | order_pprimeChar | |
card_pprimeChar: #|R| = (p ^ n)%N.
Proof. by rewrite /n -cardsT {1}(card_pgroup pprimeChar_pgroup). Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_pprimeChar | |
pprimeChar_vectAxiom: Vector.axiom n R.
Proof.
have /isog_isom/=[f /isomP[injf im_f]]: [set: R] \isog [set: 'rV['F_p]_n].
rewrite (@isog_abelem_card _ _ p) fin_Fp_lmod_abelem //=.
by rewrite !cardsT card_pprimeChar card_mx mul1n card_Fp.
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.
HB.instance Definition _ := Lmodule_hasFinDim.Build 'F_p R
pprimeChar_vectAxiom. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_vectAxiom | |
pprimeChar_dimf: \dim {: R : vectType 'F_p } = n.
Proof. by rewrite dimvf. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pprimeChar_dimf | |
Definition_ (R : finUnitRingType) pcharRp :=
FinRing.UnitRing.on (type R pcharRp).
HB.instance Definition _ (R : finComNzRingType) pcharRp :=
FinRing.ComNzRing.on (type R pcharRp).
HB.instance Definition _ (R : finComUnitRingType) pcharRp :=
FinRing.ComUnitRing.on (type R pcharRp).
HB.instance Definition _ (R : finIdomainType) pcharRp :=
FinRing.IntegralDomain.on (type R pcharRp). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
Definition_ := Finite.on F.
HB.instance Definition _ := SplittingField.copy F (finvect_type F). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Definition | |
PrimeCharType:= (pPrimeCharType) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scale instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | PrimeCharType | |
primeChar_scale:= (pprimeChar_scale) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scaleA instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scale | |
primeChar_scaleA:= (pprimeChar_scaleA) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scale1 instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scaleA | |
primeChar_scale1:= (pprimeChar_scale1) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scaleDr instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scale1 | |
primeChar_scaleDr:= (pprimeChar_scaleDr) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scaleDl instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scaleDr | |
primeChar_scaleDl:= (pprimeChar_scaleDl) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scaleAl instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scaleDl | |
primeChar_scaleAl:= (pprimeChar_scaleAl) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_scaleAr instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scaleAl | |
primeChar_scaleAr:= (pprimeChar_scaleAr) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_abelem instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_scaleAr | |
primeChar_abelem:= (pprimeChar_abelem) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_pgroup instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_abelem | |
primeChar_pgroup:= (pprimeChar_pgroup) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use order_pprimeChar instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_pgroup | |
order_primeChar:= (order_pprimeChar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use card_pprimeChar instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | order_primeChar | |
card_primeChar:= (card_pprimeChar) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_vectAxiom instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | card_primeChar | |
primeChar_vectAxiom:= (pprimeChar_vectAxiom) (only parsing).
#[deprecated(since="mathcomp 2.4.0", note="Use pprimeChar_dimf instead.")] | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_vectAxiom | |
primeChar_dimf:= (pprimeChar_dimf) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | primeChar_dimf | |
finField_galoisK 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 | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finField_galois | |
finField_galois_generatorK 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. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finField_galois_generator | |
Fermat's_little_theorem(L : fieldExtType F) (K : {subfield L}) a :
(a \in K) = (a ^+ order K == a).
Proof.
move: K a; wlog [{}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. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | Fermat's_little_theorem | |
FinSplittingFieldFor(F : finFieldType) (p : {poly F}) :
p != 0 -> {L : splittingFieldType F | splittingFieldFor 1 p^%:A {:L}}.
Proof.
have mapXsubC (f : {rmorphism _ -> _}) x:
map_poly f ('X - x%:P) = 'X - (f x)%:P.
by rewrite rmorphB /= map_polyX map_polyC.
move=> nz_p; pose splits q := {zs | q %= \prod_(z <- zs) ('X - z%:P)}.
suffices [L splitLp]: {L : fieldExtType F | splittingFieldFor 1 p^%:A {:L}}.
by exists (FinSplittingFieldType F L).
suffices [L [ys Dp]]: {L : fieldExtType F & splits L p^%:A}.
pose Lp := subvs_of <<1 & ys>>; pose toL := linfun (vsval : Lp -> L).
have [zs Dys]: {zs | map toL zs = ys}.
exists (map (vsproj _) ys); rewrite -map_comp map_id_in // => y ys_y.
by rewrite /= lfunE /= vsprojK ?seqv_sub_adjoin.
exists Lp, zs.
set lhs := (lhs in lhs %= _); set rhs := (rhs in _ %= rhs).
suffices: map_poly toL lhs %= map_poly toL rhs by rewrite eqp_map.
rewrite -Dys big_map in Dp; apply: etrans Dp; apply: congr2.
by rewrite -map_poly_comp; apply/eq_map_poly=> x; apply: rmorph_alg.
by rewrite rmorph_prod; apply/eq_bigr=> z _; apply: mapXsubC.
set Lzs := LHS; pose Lys := (toL @: Lzs)%VS; apply/vspaceP=> u.
have: val u \in Lys by rewrite /Lys aimg_adjoin_seq aimg1 Dys (valP u).
by case/memv_imgP=> v Lzs_v; rewrite memvf lfunE => /val_inj->.
move: {2}_.+1 (ltnSn (size p)) => n; elim: n => // n IHn in F p nz_p * => lbn.
have [Cp|C'p] := leqP (size p) 1.
exists F^o, [::].
by rewrite big_nil -size_poly_eq1 size_map_poly eqn_leq Cp size_
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinSplittingFieldFor | |
pPrimePowerFieldp k (m := (p ^ k)%N) :
prime p -> 0 < k -> {Fm : finFieldType | p \in [pchar Fm] & #|Fm| = m}.
Proof.
move=> pr_p k_gt0; have m_gt1: m > 1 by rewrite (ltn_exp2l 0) ?prime_gt1.
have m_gt0 := ltnW m_gt1; have m1_gt0: m.-1 > 0 by rewrite -ltnS prednK.
pose q := 'X^m - 'X; have Dq R: q R = ('X^(m.-1) - 1) * ('X - 0).
by rewrite subr0 mulrBl mul1r -exprSr prednK.
have /FinSplittingFieldFor[/= L splitLq]: q 'F_p != 0.
by rewrite Dq monic_neq0 ?rpredM ?monicXsubC ?monicXnsubC.
rewrite [_^%:A]rmorphB rmorphXn /= map_polyX -/(q L) in splitLq.
have pcharL: p \in [pchar L] by rewrite pchar_lalg pchar_Fp.
pose Fm := FinFieldExtType L; exists Fm => //.
have /finField_galois_generator[/= a _ Da]: (1 <= {:L})%VS by apply: sub1v.
pose Em := fixedSpace (a ^+ k)%g; rewrite card_Fp //= dimv1 expn1 in Da.
have{splitLq} [zs DqL defL] := splitLq.
have Uzs: uniq zs.
rewrite -separable_prod_XsubC -(eqp_separable DqL) Dq separable_root andbC.
rewrite /root !hornerE subr_eq0 eq_sym expr0n gtn_eqF ?oner_eq0 //=.
rewrite cyclotomic.separable_Xn_sub_1 // -subn1 natrB // subr_eq0.
by rewrite natrX pcharf0 // expr0n gtn_eqF // eq_sym oner_eq0.
suffices /eq_card->: Fm =i zs.
apply: succn_inj; rewrite (card_uniqP _) //= -(size_prod_XsubC _ id).
by rewrite -(eqp_size DqL) size_polyDl size_polyXn // size_polyN size_polyX.
have in_zs: zs =i Em.
move=> z; rewrite -root_prod_XsubC -(eqp_root DqL) (sameP fixedSpaceP eqP).
rewrite /root !hornerE subr_eq0 /= /m; congr (_ == z).
elim: (k) => [
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | pPrimePowerField | |
PrimePowerField:= (pPrimePowerField) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | PrimePowerField | |
finDomain_field: GRing.field_axiom R.
Proof.
move=> x /lregR-regx; apply/unitrP; exists (invF regx 1).
by split; first apply: (regx); rewrite ?mulrA f_invF // mulr1 mul1r.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finDomain_field | |
finDomain_mulrC: @commutative R R *%R.
Proof.
have fieldR := finDomain_field.
have [p p_pr pcharRp]: exists2 p, prime p & p \in [pchar 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 := pPrimeCharType pcharRp; 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 /[!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 -
... | Theorem | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | finDomain_mulrC | |
FinDomainFieldType: finFieldType :=
let cC := GRing.PzRing_hasCommutativeMul.Build R finDomain_mulrC in
let cR : comUnitRingType := HB.pack R cC in
let iC := GRing.ComUnitRing_isIntegral.Build cR domR in
let iR : finIdomainType := HB.pack cR iC in
let fC := GRing.UnitRing_isField.Build iR finDomain_field in
HB.pack iR fC. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinDomainFieldType | |
FinDomainSplittingFieldType_pcharp (pcharRp : p \in [pchar R]) :=
SplittingField.clone 'F_p R (@pPrimeCharType p FinDomainFieldType pcharRp). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinDomainSplittingFieldType_pchar | |
FinDomainSplittingFieldType:= (FinDomainSplittingFieldType_pchar) (only parsing). | Notation | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div tuple bigop prime finset fingroup",
"From mathcomp Require Import ssralg poly polydiv morphism action countalg",
"From mathcomp Require Import fina... | field/finfield.v | FinDomainSplittingFieldType | |
splittingFieldFor(U : {vspace L}) (p : {poly L}) (V : {vspace L}) :=
exists2 rs, p %= \prod_(z <- rs) ('X - z%:P) & <<U & rs>>%VS = V. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splittingFieldFor | |
splittingFieldForS(K M E : {subfield L}) p :
(K <= M)%VS -> (M <= E)%VS ->
splittingFieldFor K p E -> splittingFieldFor M p E.
Proof.
move=> sKM sKE [rs Dp genL]; exists rs => //; apply/eqP.
rewrite eqEsubv -[in X in _ && (X <= _)%VS]genL adjoin_seqSl // andbT.
by apply/Fadjoin_seqP; split; rewrite // -genL; apply: seqv_sub_adjoin.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splittingFieldForS | |
kHomU V f := ahom_in V f && (U <= fixedSpace f)%VS. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom | |
kHomP_tmp{K V f} :
reflect [/\ {in K, forall x, f x = x} &
{in V &, forall x y, f (x * y) = f x * f y}]
(kHom K V f).
Proof.
apply: (iffP andP) => [[/ahom_inP[fM _] /subvP idKf] | [idKf fM]].
by split=> // x /idKf/fixedSpaceP.
split; last by apply/subvP=> x /idKf/fixedSpaceP.
by apply/ahom_inP; split=> //; rewrite idKf ?mem1v.
Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `kHomP_tmp` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomP_tmp | |
kHomP{K V f} :
reflect [/\ {in V &, forall x y, f (x * y) = f x * f y} &
{in K, forall x, f x = x}]
(kHom K V f).
Proof. by apply: (iffP kHomP_tmp) => [][]. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomP | |
kAHomP{U V} {f : 'AEnd(L)} :
reflect {in U, forall x, f x = x} (kHom U V f).
Proof. by rewrite /kHom ahomWin; apply: fixedSpacesP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAHomP | |
kHom1U V : kHom U V \1.
Proof. by apply/kAHomP => u _; rewrite lfunE. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom1 | |
k1HomEV f : kHom 1 V f = ahom_in V f.
Proof. by apply: andb_idr => /ahom_inP[_ f1]; apply/fixedSpaceP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | k1HomE | |
kHom_monoid_morphism(f : 'End(L)) :
reflect (monoid_morphism f) (kHom 1 {:L} f).
Proof. by rewrite k1HomE; apply: ahomP_tmp. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0", deprecated(since="mathcomp 2.5.0",
note="use `kHom_monoid_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_monoid_morphism | |
kHom_lrmorphism(f : 'End(L)) : reflect (multiplicative f) (kHom 1 {:L} f).
Proof. #[warning="-deprecated-since-mathcomp-2.5.0"] by rewrite k1HomE; apply: ahomP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_lrmorphism | |
k1AHomV (f : 'AEnd(L)) : kHom 1 V f.
Proof. by rewrite k1HomE ahomWin. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | k1AHom | |
kHom_poly_idK E f p :
kHom K E f -> p \is a polyOver K -> map_poly f p = p.
Proof.
by case/kHomP_tmp=> idKf _ /polyOverP Kp; apply/polyP=> i; rewrite coef_map /= idKf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_poly_id | |
kHomSlU1 U2 V f : (U1 <= U2)%VS -> kHom U2 V f -> kHom U1 V f.
Proof. by rewrite /kHom => sU12 /andP[-> /(subv_trans sU12)]. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomSl | |
kHomSrK V1 V2 f : (V1 <= V2)%VS -> kHom K V2 f -> kHom K V1 f.
Proof. by move/subvP=> sV12 /kHomP_tmp[idKf /(sub_in2 sV12)fM]; apply/kHomP_tmp. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomSr | |
kHomSK1 K2 V1 V2 f :
(K1 <= K2)%VS -> (V1 <= V2)%VS -> kHom K2 V2 f -> kHom K1 V1 f.
Proof. by move=> sK12 sV12 /(kHomSl sK12)/(kHomSr sV12). Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomS | |
kHom_eqK E f g :
(K <= E)%VS -> {in E, f =1 g} -> kHom K E f = kHom K E g.
Proof.
move/subvP=> sKE eq_fg; wlog suffices: f g eq_fg / kHom K E f -> kHom K E g.
by move=> IH; apply/idP/idP; apply: IH => x /eq_fg.
case/kHomP_tmp=> idKf fM; apply/kHomP_tmp.
by split=> [x Kx | x y Ex Ey]; rewrite -!eq_fg ?fM ?rpredM // ?idKf ?sKE.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_eq | |
kHom_invK E f : kHom K E f -> {in E, {morph f : x / x^-1}}.
Proof.
case/kHomP_tmp=> idKf fM x Ex.
have [-> | nz_x] := eqVneq x 0; first by rewrite linear0 invr0 linear0.
have fxV: f x * f x^-1 = 1 by rewrite -fM ?rpredV ?divff // idKf ?mem1v.
have Ufx: f x \is a GRing.unit by apply/unitrPr; exists (f x^-1).
by apply: (mulrI Ufx); rewrite divrr.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_inv | |
kHom_dimK E f : kHom K E f -> \dim (f @: E) = \dim E.
Proof.
move=> homKf; have [idKf fM] := kHomP_tmp homKf.
apply/limg_dim_eq/eqP; rewrite -subv0; apply/subvP=> v.
rewrite memv_cap memv0 memv_ker => /andP[Ev]; apply: contraLR => nz_v.
by rewrite -unitfE unitrE -(kHom_inv homKf) // -fM ?rpredV ?divff ?idKf ?mem1v.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_dim | |
kHom_is_zmod_morphism: kHom K E f -> zmod_morphism kHomf.
Proof. by case/kHomP_tmp => idKf fM; apply: raddfB. Qed.
#[deprecated(since="mathcomp 2.5.0", note="use `kHom_is_zmod_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_is_zmod_morphism | |
kHom_is_additive:= kHom_is_zmod_morphism. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_is_additive | |
kHom_is_monoid_morphism: kHom K E f -> monoid_morphism kHomf.
Proof.
case/kHomP_tmp=> idKf fM; rewrite /kHomf.
by split=> [|a b] /=; [rewrite algid1 idKf // mem1v | rewrite /= fM ?subvsP].
Qed.
#[deprecated(since="mathcomp 2.5.0", note="use `kHom_is_monoid_morphism` instead")] | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_is_monoid_morphism | |
kHom_is_multiplicative:=
(fun p => (p.1, p.2)) \o kHom_is_monoid_morphism.
Variable (homKEf : kHom K E f).
HB.instance Definition _ :=
@GRing.isZmodMorphism.Build _ _ kHomf (kHom_is_zmod_morphism homKEf).
HB.instance Definition _ :=
@GRing.isMonoidMorphism.Build _ _ kHomf (kHom_is_monoid_morphism homKEf). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_is_multiplicative | |
kHom_rmorphism:= Eval hnf in (kHomf : {rmorphism _ -> _}). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_rmorphism | |
kHom_hornerK E f p x :
kHom K E f -> p \is a polyOver E -> x \in E -> f p.[x] = (map_poly f p).[f x].
Proof.
move=> homKf /polyOver_subvs[{}p -> Ex]; pose fRM := kHom_rmorphism homKf.
by rewrite (horner_map _ _ (Subvs Ex)) -[f _](horner_map fRM) map_poly_comp.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_horner | |
kHom_rootK E f p x :
kHom K E f -> p \is a polyOver E -> x \in E -> root p x ->
root (map_poly f p) (f x).
Proof.
by move/kHom_horner=> homKf Ep Ex /rootP px0; rewrite /root -homKf ?px0 ?raddf0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_root | |
kHom_root_idK E f p x :
(K <= E)%VS -> kHom K E f -> p \is a polyOver K -> x \in E -> root p x ->
root p (f x).
Proof.
move=> sKE homKf Kp Ex /(kHom_root homKf (polyOverSv sKE Kp) Ex).
by rewrite (kHom_poly_id homKf).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_root_id | |
Definition_ := @GRing.isZmodMorphism.Build _ _ kHomf
kHomExtend_zmod_morphism_subproof.
HB.instance Definition _ := @GRing.isScalable.Build _ _ _ _ kHomf
kHomExtend_scalable_subproof.
Let kHomExtendLinear := Eval hnf in (kHomf : {linear _ -> _}). | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | Definition | |
kHomExtend:= linfun kHomExtendLinear. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtend | |
kHomExtendEz : kHomExtend z = (map_poly f (Fadjoin_poly E x z)).[y].
Proof. by rewrite lfunE. Qed.
Hypotheses (sKE : (K <= E)%VS) (homKf : kHom K E f).
Local Notation Px := (minPoly E x).
Hypothesis fPx_y_0 : root (map_poly f Px) y. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtendE | |
kHomExtend_idz : z \in E -> kHomExtend z = f z.
Proof. by move=> Ez; rewrite kHomExtendE Fadjoin_polyC ?map_polyC ?hornerC. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtend_id | |
kHomExtend_val: kHomExtend x = y.
Proof.
have fX: map_poly f 'X = 'X by rewrite (kHom_poly_id homKf) ?polyOverX.
have [Ex | E'x] := boolP (x \in E); last first.
by rewrite kHomExtendE Fadjoin_polyX // fX hornerX.
have:= fPx_y_0; rewrite (minPoly_XsubC Ex) raddfB /= map_polyC fX root_XsubC /=.
by rewrite (kHomExtend_id Ex) => /eqP->.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtend_val | |
kHomExtend_polyp :
p \in polyOver E -> kHomExtend p.[x] = (map_poly f p).[y].
Proof.
move=> Ep; rewrite kHomExtendE (Fadjoin_poly_mod x) //.
rewrite (divp_eq (map_poly f p) (map_poly f Px)).
rewrite !hornerE (rootP fPx_y_0) mulr0 add0r.
have [p1 ->] := polyOver_subvs Ep.
have [Px1 ->] := polyOver_subvs (minPolyOver E x).
by rewrite -map_modp -!map_poly_comp (map_modp (kHom_rmorphism homKf)).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtend_poly | |
kHomExtendP: kHom K <<E; x>> kHomExtend.
Proof.
have [idKf fM] := kHomP_tmp homKf.
apply/kHomP_tmp; split=> [z Kz|]; first by rewrite kHomExtend_id ?(subvP sKE) ?idKf.
move=> _ _ /Fadjoin_polyP[p Ep ->] /Fadjoin_polyP[q Eq ->].
rewrite -hornerM !kHomExtend_poly ?rpredM // -hornerM; congr _.[_].
apply/polyP=> i; rewrite coef_map !coefM /= linear_sum /=.
by apply: eq_bigr => j _; rewrite !coef_map /= fM ?(polyOverP _).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHomExtendP | |
kAutU V f := kHom U V f && (f @: V == V)%VS. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAut | |
kAutEK E f : kAut K E f = kHom K E f && (f @: E <= E)%VS.
Proof.
apply/andP/andP=> [[-> /eqP->] // | [homKf EfE]].
by rewrite eqEdim EfE /= (kHom_dim homKf).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAutE | |
kAutSU1 U2 V f : (U1 <= U2)%VS -> kAut U2 V f -> kAut U1 V f.
Proof. by move=> sU12 /andP[/(kHomSl sU12)homU1f EfE]; apply/andP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAutS | |
kHom_kAut_subK E f : kAut K E f -> kHom K E f. Proof. by case/andP. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_kAut_sub | |
kAut_eqK E (f g : 'End(L)) :
(K <= E)%VS -> {in E, f =1 g} -> kAut K E f = kAut K E g.
Proof.
by move=> sKE eq_fg; rewrite !kAutE (kHom_eq sKE eq_fg) (eq_in_limg eq_fg).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAut_eq | |
kAutfEK f : kAut K {:L} f = kHom K {:L} f.
Proof. by rewrite kAutE subvf andbT. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAutfE | |
kAut1EE (f : 'AEnd(L)) : kAut 1 E f = (f @: E <= E)%VS.
Proof. by rewrite kAutE k1AHom. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAut1E | |
kAutf_lker0K f : kHom K {:L} f -> lker f == 0%VS.
Proof.
move/(kHomSl (sub1v _))/kHom_monoid_morphism => fM.
pose fmM := GRing.isMonoidMorphism.Build _ _ _ fM.
pose fRM : {rmorphism _ -> _} := HB.pack (fun_of_lfun f) fmM.
by apply/lker0P; apply: (fmorph_inj fRM).
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAutf_lker0 | |
inv_kHomfK f : kHom K {:L} f -> kHom K {:L} f^-1.
Proof.
move=> homKf; have [[idKf fM] kerf0] := (kHomP_tmp homKf, kAutf_lker0 homKf).
have f1K: cancel f^-1%VF f by apply: lker0_lfunVK.
apply/kHomP_tmp; split=> [x Kx | x y _ _]; apply: (lker0P kerf0).
by rewrite f1K idKf.
by rewrite fM ?memvf ?{1}f1K.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | inv_kHomf | |
inv_is_ahom(f : 'AEnd(L)) : ahom_in {:L} f^-1.
Proof.
have /ahomP_tmp/kHom_monoid_morphism hom1f := valP f.
exact/ahomP_tmp/kHom_monoid_morphism/inv_kHomf.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | inv_is_ahom | |
inv_ahom(f : 'AEnd(L)) : 'AEnd(L) := AHom (inv_is_ahom f). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | inv_ahom | |
comp_kHom_imgK E f g :
kHom K (g @: E) f -> kHom K E g -> kHom K E (f \o g).
Proof.
move=> /kHomP_tmp[idKf fM] /kHomP_tmp[idKg gM]; apply/kHomP_tmp; split=> [x Kx | x y Ex Ey].
by rewrite lfunE /= idKg ?idKf.
by rewrite !lfunE /= gM // fM ?memv_img.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | comp_kHom_img | |
comp_kHomK E f g : kHom K {:L} f -> kHom K E g -> kHom K E (f \o g).
Proof. by move/(kHomSr (subvf (g @: E))); apply: comp_kHom_img. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | comp_kHom | |
kHom_extendsK E f p U :
(K <= E)%VS -> kHom K E f ->
p \is a polyOver K -> splittingFieldFor E p U ->
{g | kHom K U g & {in E, f =1 g}}.
Proof.
move=> sKE homEf Kp /sig2_eqW[rs Dp <-{U}].
set r := rs; have rs_r: all [in rs] r by apply/allP.
elim: r rs_r => [_|z r IHr /=/andP[rs_z rs_r]] /= in E f sKE homEf *.
by exists f; rewrite ?Fadjoin_nil.
set Ez := <<E; z>>%AS; pose fpEz := map_poly f (minPoly E z).
suffices{IHr} /sigW[y fpEz_y]: exists y, root fpEz y.
have homEz_fz: kHom K Ez (kHomExtend E f z y) by apply: kHomExtendP.
have sKEz: (K <= Ez)%VS := subv_trans sKE (subv_adjoin E z).
have [g homGg Dg] := IHr rs_r _ _ sKEz homEz_fz.
exists g => [|x Ex]; first by rewrite adjoin_cons.
by rewrite -Dg ?subvP_adjoin // kHomExtend_id.
have [m DfpEz]: {m | fpEz %= \prod_(w <- mask m rs) ('X - w%:P)}.
apply: dvdp_prod_XsubC; rewrite -(eqp_dvdr _ Dp) -(kHom_poly_id homEf Kp).
have /polyOver_subvs[q Dq] := polyOverSv sKE Kp.
have /polyOver_subvs[qz Dqz] := minPolyOver E z.
rewrite /fpEz Dq Dqz -2?{1}map_poly_comp (dvdp_map (kHom_rmorphism homEf)).
rewrite -(dvdp_map (@vsval _ _ E)) -Dqz -Dq.
by rewrite minPoly_dvdp ?(polyOverSv sKE) // (eqp_root Dp) root_prod_XsubC.
exists (mask m rs)`_0; rewrite (eqp_root DfpEz) root_prod_XsubC mem_nth //.
rewrite -ltnS -(size_prod_XsubC _ id) -(eqp_size DfpEz).
rewrite size_poly_eq -?lead_coefE ?size_minPoly // (monicP (monic_minPoly E z)).
by have [idKf _] := kHomP_tmp homEf; rewrite idKf ?mem1v ?oner_eq0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_extends | |
splitting_field_axiom(F : fieldType) (L : fieldExtType F) :=
exists2 p : {poly L}, p \is a polyOver 1%VS & splittingFieldFor 1 p {:L}.
HB.mixin Record FieldExt_isSplittingField (F : fieldType) L of FieldExt F L := {
splittingFieldP_subproof : splitting_field_axiom L
}.
#[mathcomp(axiom="splitting_field_axiom"), short(type="splittingFieldType")]
HB.structure Definition SplittingField F :=
{ T of FieldExt_isSplittingField F T & FieldExt F T }. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splitting_field_axiom | |
normal_field_splitting(F : fieldType) (L : fieldExtType F) :
(forall (K : {subfield L}) x,
exists r, minPoly K x == \prod_(y <- r) ('X - y%:P)) ->
SplittingField.axiom L.
Proof.
move=> normalL; pose r i := sval (sigW (normalL 1%AS (tnth (vbasis {:L}) i))).
have sz_r i: size (r i) <= \dim {:L}.
rewrite -ltnS -(size_prod_XsubC _ id) /r; case: sigW => _ /= /eqP <-.
rewrite size_minPoly ltnS; move: (tnth _ _) => x.
by rewrite adjoin_degreeE dimv1 divn1 dimvS // subvf.
pose mkf (z : L) := 'X - z%:P.
exists (\prod_i \prod_(j < \dim {:L} | j < size (r i)) mkf (r i)`_j).
apply: rpred_prod => i _; rewrite big_ord_narrow /= /r; case: sigW => rs /=.
by rewrite (big_nth 0) big_mkord => /eqP <- {rs}; apply: minPolyOver.
rewrite pair_big_dep /= -big_filter -(big_map _ xpredT mkf).
set rF := map _ _; exists rF; first exact: eqpxx.
apply/eqP; rewrite eqEsubv subvf -(span_basis (vbasisP {:L})).
apply/span_subvP=> _ /tnthP[i ->]; set x := tnth _ i.
have /tnthP[j ->]: x \in in_tuple (r i).
by rewrite -root_prod_XsubC /r; case: sigW => _ /=/eqP<-; apply: root_minPoly.
apply/seqv_sub_adjoin/mapP; rewrite (tnth_nth 0).
exists (i, widen_ord (sz_r i) j) => //.
by rewrite mem_filter /= ltn_ord mem_index_enum.
Qed.
HB.factory Record FieldExt_isNormalSplittingField
(F : fieldType) L of FieldExt F L := {
normal_field_splitting_axiom : forall (K : {subfield L}) x,
exists r, minPoly K x == \prod_(y <- r) ('X - y%:P)
}.
HB.builders Context F L of FieldExt_isNormalSplittingField F L.
HB.instance Def
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | normal_field_splitting | |
splittingFieldP: SplittingField.axiom L.
Proof. exact: splittingFieldP_subproof. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splittingFieldP | |
splittingPoly:
{p : {poly L} | p \is a polyOver 1%VS & splittingFieldFor 1 p {:L}}.
Proof.
pose factF p s := (p \is a polyOver 1%VS) && (p %= \prod_(z <- s) ('X - z%:P)).
suffices [[p rs] /andP[]]: {ps | factF F L ps.1 ps.2 & <<1 & ps.2>> = {:L}}%VS.
by exists p; last exists rs.
apply: sig2_eqW; have [p F0p [rs splitLp genLrs]] := splittingFieldP.
by exists (p, rs); rewrite // /factF F0p splitLp.
Qed.
Fact fieldOver_splitting E : SplittingField.axiom (fieldOver E).
Proof.
have [p Fp [r Dp defL]] := splittingFieldP; exists p.
apply/polyOverP=> j; rewrite trivial_fieldOver.
by rewrite (subvP (sub1v E)) ?(polyOverP Fp).
exists r => //; apply/vspaceP=> x; rewrite memvf.
have [L0 [_ _ defL0]] := @aspaceOverP _ _ E <<1 & r : seq (fieldOver E)>>.
rewrite defL0; have: x \in <<1 & r>>%VS by rewrite defL (@memvf _ L).
apply: subvP; apply/Fadjoin_seqP; rewrite -memvE -defL0 mem1v.
by split=> // y r_y; rewrite -defL0 seqv_sub_adjoin.
Qed.
HB.instance Definition _ E := FieldExt_isSplittingField.Build
(subvs_of E) (fieldOver E) (fieldOver_splitting E). | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splittingPoly | |
enum_AEnd: {kAutL : seq 'AEnd(L) | forall f, f \in kAutL}.
Proof.
pose isAutL (s : seq 'AEnd(L)) (f : 'AEnd(L)) := kHom 1 {:L} f = (f \in s).
suffices [kAutL in_kAutL] : {kAutL : seq 'AEnd(L) | forall f, isAutL kAutL f}.
by exists kAutL => f; rewrite -in_kAutL k1AHom.
have [p Kp /sig2_eqW[rs Dp defL]] := splittingPoly.
do [rewrite {}/isAutL -(erefl (asval 1)); set r := rs; set E := 1%AS] in defL *.
have [sKE rs_r]: (1 <= E)%VS /\ all [in rs] r by split; last apply/allP.
elim: r rs_r => [_|z r IHr /=/andP[rs_z rs_r]] /= in (E) sKE defL *.
rewrite Fadjoin_nil in defL; exists [tuple \1%AF] => f; rewrite defL inE.
apply/idP/eqP=> [/kAHomP f1 | ->]; last exact: kHom1.
by apply/val_inj/lfunP=> x; rewrite id_lfunE f1 ?memvf.
do [set Ez := <<E; z>>%VS; rewrite adjoin_cons] in defL.
have sEEz: (E <= Ez)%VS := subv_adjoin E z; have sKEz := subv_trans sKE sEEz.
have{IHr} [homEz DhomEz] := IHr rs_r _ sKEz defL.
have Ep: p \in polyOver E := polyOverSv sKE Kp.
have{rs_z} pz0: root p z by rewrite (eqp_root Dp) root_prod_XsubC.
pose pEz := minPoly E z; pose n := \dim_E Ez.
have{pz0} [rz DpEz]: {rz : n.-tuple L | pEz %= \prod_(w <- rz) ('X - w%:P)}.
have /dvdp_prod_XsubC[m DpEz]: pEz %| \prod_(w <- rs) ('X - w%:P).
by rewrite -(eqp_dvdr _ Dp) minPoly_dvdp ?(polyOverSv sKE).
suffices sz_rz: size (mask m rs) == n by exists (Tuple sz_rz).
rewrite -[n]adjoin_degreeE -eqSS -size_minPoly.
by rewrite (eqp_size DpEz) size_prod_XsubC.
have fEz i (y := tnth rz i): {f : 'AEnd(L) | kHom E {:L}
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | enum_AEnd | |
splitting_field_normalK x :
exists r, minPoly K x == \prod_(y <- r) ('X - y%:P).
Proof.
pose q1 := minPoly 1 x; pose fx_root q (f : 'AEnd(L)) := root q (f x).
have [[p F0p splitLp] [autL DautL]] := (splittingFieldP, enum_AEnd).
suffices{K} autL_px q: q != 0 -> q %| q1 -> size q > 1 -> has (fx_root q) autL.
set q := minPoly K x; have: q \is monic := monic_minPoly K x.
have: q %| q1 by rewrite minPolyS // sub1v.
have [d] := ubnP (size q); elim: d q => // d IHd q leqd q_dv_q1 mon_q.
have nz_q: q != 0 := monic_neq0 mon_q.
have [|q_gt1|q_1] := ltngtP (size q) 1; last first; last by rewrite polySpred.
by exists nil; rewrite big_nil -eqp_monic ?monic1 // -size_poly_eq1 q_1.
have /hasP[f autLf /factor_theorem[q2 Dq]] := autL_px q nz_q q_dv_q1 q_gt1.
have mon_q2: q2 \is monic by rewrite -(monicMr _ (monicXsubC (f x))) -Dq.
rewrite Dq size_monicM -?size_poly_eq0 ?size_XsubC ?addn2 //= ltnS in leqd.
have q2_dv_q1: q2 %| q1 by rewrite (dvdp_trans _ q_dv_q1) // Dq dvdp_mulr.
rewrite Dq; have [r /eqP->] := IHd q2 leqd q2_dv_q1 mon_q2.
by exists (f x :: r); rewrite big_cons mulrC.
have [d] := ubnP (size q); elim: d q => // d IHd q leqd nz_q q_dv_q1 q_gt1.
without loss{d leqd IHd nz_q q_gt1} irr_q: q q_dv_q1 / irreducible_poly q.
move=> IHq; apply: wlog_neg => not_autLx_q; apply: IHq => //.
split=> // q2 q2_neq1 q2_dv_q; rewrite -dvdp_size_eqp // eqn_leq dvdp_leq //=.
rewrite leqNgt; apply: contra not_autLx_q => ltq2q.
have nz_q2: q2 != 0 by apply: contraTneq q2_dv_q => ->; rew
... | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | splitting_field_normal | |
kHom_to_AEndK E f : kHom K E f -> {g : 'AEnd(L) | {in E, f =1 val g}}.
Proof.
move=> homKf; have{homKf} [homFf sFE] := (kHomSl (sub1v K) homKf, sub1v E).
have [p Fp /(splittingFieldForS sFE (subvf E))splitLp] := splittingPoly.
have [g0 homLg0 eq_fg] := kHom_extends sFE homFf Fp splitLp.
by apply: exist (Sub g0 _) _ => //; apply/ahomP_tmp/kHom_monoid_morphism.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kHom_to_AEnd | |
inAEndf := SeqSub (svalP (enum_AEnd L) f).
Fact inAEndK : cancel inAEnd val. Proof. by []. Qed.
HB.instance Definition _ := Countable.copy 'AEnd(L) (can_type inAEndK).
HB.instance Definition _ := isFinite.Build 'AEnd(L)
(pcan_enumP (can_pcan inAEndK)). | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | inAEnd | |
comp_AEnd(f g : 'AEnd(L)) : 'AEnd(L) := (g \o f)%AF.
Fact comp_AEndA : associative comp_AEnd.
Proof. by move=> f g h; apply: val_inj; symmetry; apply: comp_lfunA. Qed.
Fact comp_AEnd1l : left_id \1%AF comp_AEnd.
Proof. by move=> f; apply/val_inj/comp_lfun1r. Qed.
Fact comp_AEndK : left_inverse \1%AF (@inv_ahom _ L) comp_AEnd.
Proof. by move=> f; apply/val_inj; rewrite /= lker0_compfV ?AEnd_lker0. Qed.
HB.instance Definition _:= Finite_isGroup.Build 'AEnd(L)
comp_AEndA comp_AEnd1l comp_AEndK. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | comp_AEnd | |
kAEndU V := [set f : 'AEnd(L) | kAut U V f]. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEnd | |
kAEndfU := kAEnd U {:L}. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEndf | |
kAEnd_group_setK E : group_set (kAEnd K E).
Proof.
apply/group_setP; split=> [|f g]; first by rewrite inE /kAut kHom1 lim1g eqxx.
rewrite !inE !kAutE => /andP[homKf EfE] /andP[/(kHomSr EfE)homKg EgE].
by rewrite (comp_kHom_img homKg homKf) limg_comp (subv_trans _ EgE) ?limgS.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEnd_group_set | |
kAEnd_groupK E := group (kAEnd_group_set K E). | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEnd_group | |
kAEndf_groupK := [group of kAEndf K]. | Canonical | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEndf_group | |
kAEnd_normK E : kAEnd K E \subset 'N(kAEndf E)%g.
Proof.
apply/subsetP=> x; rewrite -groupV 2!in_set => /andP[_ /eqP ExE].
apply/subsetP=> _ /imsetP[y homEy ->]; rewrite !in_set !kAutfE in homEy *.
apply/kAHomP=> u Eu; have idEy := kAHomP homEy; rewrite -ExE in idEy.
rewrite !(@lfunE _ _ L) /= (@lfunE _ _ L) /= idEy ?memv_img //.
by rewrite lker0_lfunVK ?AEnd_lker0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | kAEnd_norm | |
mem_kAut_cosetK E (g : 'AEnd(L)) :
kAut K E g -> g \in coset (kAEndf E) g.
Proof.
move=> autEg; rewrite val_coset ?rcoset_refl //.
by rewrite (subsetP (kAEnd_norm K E)) // inE.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | mem_kAut_coset | |
aut_mem_eqPE (x y : coset_of (kAEndf E)) f g :
f \in x -> g \in y -> reflect {in E, f =1 g} (x == y).
Proof.
move=> x_f y_g; rewrite -(coset_mem x_f) -(coset_mem y_g).
have [Nf Ng] := (subsetP (coset_norm x) f x_f, subsetP (coset_norm y) g y_g).
rewrite (sameP eqP (rcoset_kercosetP Nf Ng)) mem_rcoset inE kAutfE.
apply: (iffP kAHomP) => idEfg u Eu.
by rewrite -(mulgKV g f) lfunE /= idEfg.
by rewrite (@lfunE _ _ L) /= idEfg // lker0_lfunK ?AEnd_lker0.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop ssralg poly",
"From mathcomp Require Import polydiv finset fingroup morphism quotient perm",
"From mathcomp Require Import actio... | field/galois.v | aut_mem_eqP |
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