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) pch... | 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.
m... | 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 : 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 | |
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 sK... | 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 (sub... | 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 <- ... | 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)... | 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]... | 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.pa... | 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; appl... | 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_i... | 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 // ... | 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: (... | 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_i... | 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... | 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... | 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 *... | 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"), sho... | 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 -... | 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 spl... | 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]] := splittingPol... | 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 :=... | 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 _) _ => ... | 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) com... | 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 ?m... | 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.
appl... | 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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