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 |
|---|---|---|---|---|---|---|
gal_of:= Gal of [subg kAEnd_group 1 <<V>> / kAEndf (agenv V)]. | Inductive | 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 | gal_of | |
gal(f : 'AEnd(L)) := Gal (subg _ (coset _ 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 | gal | |
gal_sgvalx := let: Gal u := x in u.
Fact gal_sgvalK : cancel gal_sgval Gal. Proof. by case. Qed.
Let gal_sgval_inj := can_inj gal_sgvalK.
HB.instance Definition _ := Countable.copy gal_of (can_type gal_sgvalK).
HB.instance Definition _ := isFinite.Build gal_of
(pcan_enumP (can_pcan gal_sgvalK)). | 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 | gal_sgval | |
gal_one:= Gal 1%g. | 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 | gal_one | |
gal_invx := Gal (gal_sgval x)^-1. | 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 | gal_inv | |
gal_mulx y := Gal (gal_sgval x * gal_sgval y).
Fact gal_oneP : left_id gal_one gal_mul.
Proof. by move=> x; apply/gal_sgval_inj/mul1g. Qed.
Fact gal_invP : left_inverse gal_one gal_inv gal_mul.
Proof. by move=> x; apply/gal_sgval_inj/mulVg. Qed.
Fact gal_mulP : associative gal_mul.
Proof. by move=> x y z; apply/gal_sgval_inj/mulgA. Qed.
HB.instance Definition _ := Finite_isGroup.Build gal_of gal_mulP gal_oneP gal_invP. | 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 | gal_mul | |
gal_repru : 'AEnd(L) := repr (sgval (gal_sgval u)).
Fact gal_is_morphism : {in kAEnd 1 (agenv V) &, {morph gal : x y / x * y}%g}.
Proof.
move=> f g /= autEa autEb; congr (Gal _).
by rewrite !morphM ?mem_morphim // (subsetP (kAEnd_norm 1 _)).
Qed. | Coercion | 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 | gal_repr | |
gal_morphism:= Morphism gal_is_morphism. | 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 | gal_morphism | |
gal_reprK: cancel gal_repr gal.
Proof. by case=> x; rewrite /gal coset_reprK sgvalK. 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 | gal_reprK | |
gal_repr_inj: injective gal_repr.
Proof. exact: can_inj gal_reprK. 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 | gal_repr_inj | |
gal_AEndx : gal_repr x \in kAEnd 1 (agenv V).
Proof.
rewrite /gal_repr; case/gal_sgval: x => _ /=/morphimP[g Ng autEg ->].
rewrite val_coset //=; case: repr_rcosetP => f; rewrite groupMr // !inE kAut1E.
by rewrite kAutE -andbA => /and3P[_ /fixedSpace_limg-> _].
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 | gal_AEnd | |
gal_eqPE {x y : gal_of E} : reflect {in E, x =1 y} (x == y).
Proof.
by rewrite -{1}(subfield_closed E); apply: aut_mem_eqP; apply: mem_repr_coset.
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 | gal_eqP | |
galKE (f : 'AEnd(L)) : (f @: E <= E)%VS -> {in E, gal E f =1 f}.
Proof.
rewrite -kAut1E -{1 2}(subfield_closed E) => autEf.
apply: (aut_mem_eqP (mem_repr_coset _) _ (eqxx _)).
by rewrite subgK /= ?(mem_kAut_coset autEf) // ?mem_quotient ?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 | galK | |
eq_galPE (f g : 'AEnd(L)) :
(f @: E <= E)%VS -> (g @: E <= E)%VS ->
reflect {in E, f =1 g} (gal E f == gal E g).
Proof.
move=> EfE EgE.
by apply: (iffP gal_eqP) => Dfg a Ea; have:= Dfg a Ea; rewrite !{1}galK.
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 | eq_galP | |
limg_galE (x : gal_of E) : (x @: E)%VS = E.
Proof. by have:= gal_AEnd x; rewrite inE subfield_closed => /andP[_ /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 | limg_gal | |
memv_galE (x : gal_of E) a : a \in E -> x a \in E.
Proof. by move/(memv_img x); rewrite limg_gal. 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 | memv_gal | |
gal_idE a : (1 : gal_of E)%g a = a.
Proof. by rewrite /gal_repr repr_coset1 id_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 | gal_id | |
galME (x y : gal_of E) a : a \in E -> (x * y)%g a = y (x a).
Proof.
rewrite /= -comp_lfunE; apply/eq_galP; rewrite ?limg_comp ?limg_gal //.
by rewrite morphM /= ?gal_reprK ?gal_AEnd.
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 | galM | |
galVE (x : gal_of E) : {in E, (x^-1)%g =1 x^-1%VF}.
Proof.
move=> a Ea; apply: canRL (lker0_lfunK (AEnd_lker0 _)) _.
by rewrite -galM // mulVg gal_id.
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 | galV | |
galoisGV U := gal V @* <<kAEnd (U :&: V) V>>.
Local Notation "''Gal' ( V / U )" := (galoisG V U) : group_scope. | 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 | galoisG | |
galoisG_groupE U := Eval hnf in [group of (galoisG E U)].
Local Notation "''Gal' ( V / U )" := (galoisG_group V U) : Group_scope. | 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 | galoisG_group | |
gal_capU V : 'Gal(V / U) = 'Gal(V / U :&: V).
Proof. by rewrite /galoisG -capvA capvv. 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 | gal_cap | |
gal_kAutK E x : (K <= E)%VS -> (x \in 'Gal(E / K)) = kAut K E x.
Proof.
move=> sKE; apply/morphimP/idP=> /= [[g EgE KautEg ->{x}] | KautEx].
rewrite genGid !inE kAut1E /= subfield_closed (capv_idPl sKE) in KautEg EgE.
by apply: etrans KautEg; apply/(kAut_eq sKE); apply: galK.
exists (x : 'AEnd(L)); rewrite ?gal_reprK ?gal_AEnd //.
by rewrite (capv_idPl sKE) mem_gen ?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 | gal_kAut | |
gal_kHomK E x : (K <= E)%VS -> (x \in 'Gal(E / K)) = kHom K E x.
Proof. by move/gal_kAut->; rewrite /kAut limg_gal eqxx 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 | gal_kHom | |
kAut_to_galK E f :
kAut K E f -> {x : gal_of E | x \in 'Gal(E / K) & {in E, f =1 x}}.
Proof.
case/andP=> homKf EfE; have [g Df] := kHom_to_AEnd homKf.
have{homKf EfE} autEg: kAut (K :&: E) E g.
rewrite /kAut -(kHom_eq (capvSr _ _) Df) (kHomSl (capvSl _ _) homKf) /=.
by rewrite -(eq_in_limg Df).
have FautEg := kAutS (sub1v _) autEg.
exists (gal E g) => [|a Ea]; last by rewrite {f}Df // galK // -kAut1E.
by rewrite mem_morphim /= ?subfield_closed ?genGid ?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 | kAut_to_gal | |
fixed_galK E x a :
(K <= E)%VS -> x \in 'Gal(E / K) -> a \in K -> x a = a.
Proof. by move/gal_kHom=> -> /kAHomP idKx /idKx. 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 | fixed_gal | |
fixedPoly_galK E x p :
(K <= E)%VS -> x \in 'Gal(E / K) -> p \is a polyOver K -> map_poly x p = p.
Proof.
move=> sKE galEKx /polyOverP Kp; apply/polyP => i.
by rewrite coef_map /= (fixed_gal 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 | fixedPoly_gal | |
root_minPoly_galK E x a :
(K <= E)%VS -> x \in 'Gal(E / K) -> a \in E -> root (minPoly K a) (x a).
Proof.
move=> sKE galEKx Ea; have homKx: kHom K E x by rewrite -gal_kHom.
have K_Pa := minPolyOver K a; rewrite -[minPoly K a](fixedPoly_gal _ galEKx) //.
by rewrite (kHom_root homKx) ?root_minPoly // (polyOverS (subvP 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 | root_minPoly_gal | |
gal_adjoin_eqK a x y :
x \in 'Gal(<<K; a>> / K) -> y \in 'Gal(<<K; a>> / K) ->
(x == y) = (x a == y a).
Proof.
move=> galKa_x galKa_y; apply/idP/eqP=> [/eqP-> // | eq_xy_a].
apply/gal_eqP => _ /Fadjoin_polyP[p Kp ->].
by rewrite -!horner_map !(fixedPoly_gal (subv_adjoin K a)) //= eq_xy_a.
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 | gal_adjoin_eq | |
galSK M E : (K <= M)%VS -> 'Gal(E / M) \subset 'Gal(E / K).
Proof.
rewrite gal_cap (gal_cap K E) => sKM; apply/subsetP=> x.
by rewrite !gal_kAut ?capvSr //; apply: kAutS; apply: capvS.
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 | galS | |
gal_conjgK E x : 'Gal(E / K) :^ x = 'Gal(E / x @: K).
Proof.
without loss sKE: K / (K <= E)%VS.
move=> IH_K; rewrite gal_cap {}IH_K ?capvSr //.
transitivity 'Gal(E / x @: K :&: x @: E); last by rewrite limg_gal -gal_cap.
congr 'Gal(E / _); apply/eqP; rewrite eqEsubv limg_cap; apply/subvP=> a.
rewrite memv_cap => /andP[/memv_imgP[b Kb ->] /memv_imgP[c Ec] eq_bc].
by rewrite memv_img // memv_cap Kb (lker0P (AEnd_lker0 _) _ _ eq_bc).
wlog suffices IHx: x K sKE / 'Gal(E / K) :^ x \subset 'Gal(E / x @: K).
apply/eqP; rewrite eqEsubset IHx // -sub_conjgV (subset_trans (IHx _ _ _)) //.
by apply/subvP=> _ /memv_imgP[a Ka ->]; rewrite memv_gal ?(subvP sKE).
rewrite -limg_comp (etrans (eq_in_limg _) (lim1g _)) // => a /(subvP sKE)Ka.
by rewrite !(@lfunE _ _ L) /= -galM // mulgV gal_id.
apply/subsetP=> _ /imsetP[y galEy ->]; rewrite gal_cap gal_kHom ?capvSr //=.
apply/kAHomP=> _ /memv_capP[/memv_imgP[a Ka ->] _]; have Ea := subvP sKE a Ka.
by rewrite -galM // -conjgC galM // (fixed_gal sKE galEy).
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 | gal_conjg | |
fixedFieldV (A : {set gal_of V}) :=
(V :&: \bigcap_(x in A) fixedSpace x)%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 | fixedField | |
fixedFieldPE {A : {set gal_of E}} a :
a \in E -> reflect (forall x, x \in A -> x a = a) (a \in fixedField A).
Proof.
by rewrite memv_cap => ->; apply: (iffP subv_bigcapP) => cAa x /cAa/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 | fixedFieldP | |
mem_fixedFieldPE (A : {set gal_of E}) a :
a \in fixedField A -> a \in E /\ (forall x, x \in A -> x a = a).
Proof.
by move=> fixAa; have [Ea _] := memv_capP fixAa; have:= fixedFieldP Ea fixAa.
Qed.
Fact fixedField_is_aspace E (A : {set gal_of E}) : is_aspace (fixedField A).
Proof.
rewrite /fixedField; elim/big_rec: _ {1}E => [|x K _ IH_K] M.
exact: (valP (M :&: _)%AS).
by rewrite capvA IH_K.
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_fixedFieldP | |
fixedField_aspaceE A : {subfield L} :=
ASpace (@fixedField_is_aspace E A). | 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 | fixedField_aspace | |
fixedField_boundE (A : {set gal_of E}) : (fixedField A <= E)%VS.
Proof. exact: capvSl. 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 | fixedField_bound | |
fixedFieldSE (A B : {set gal_of E}) :
A \subset B -> (fixedField B <= fixedField A)%VS.
Proof.
move/subsetP=> sAB; apply/subvP => a /mem_fixedFieldP[Ea cBa].
by apply/fixedFieldP; last apply: sub_in1 cBa.
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 | fixedFieldS | |
galois_connection_subvK E :
(K <= E)%VS -> (K <= fixedField ('Gal(E / K)))%VS.
Proof.
move=> sKE; apply/subvP => a Ka; have Ea := subvP sKE a Ka.
by apply/fixedFieldP=> // x galEx; apply: (fixed_gal 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 | galois_connection_subv | |
galois_connection_subsetE (A : {set gal_of E}):
A \subset 'Gal(E / fixedField A).
Proof.
apply/subsetP => x Ax; rewrite gal_kAut ?capvSl // kAutE limg_gal subvv andbT.
by apply/kAHomP=> a /mem_fixedFieldP[_ ->].
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 | galois_connection_subset | |
galois_connectionK E (A : {set gal_of E}):
(K <= E)%VS -> (A \subset 'Gal(E / K)) = (K <= fixedField A)%VS.
Proof.
move=> sKE; apply/idP/idP => [/fixedFieldS | /(galS E)].
exact/subv_trans/galois_connection_subv.
exact/subset_trans/galois_connection_subset.
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 | galois_connection | |
galTraceU V a := \sum_(x in 'Gal(V / U)) (x a). | 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 | galTrace | |
galNormU V a := \prod_(x in 'Gal(V / U)) (x a). | 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 | galNorm | |
galTrace_is_additive:= galTrace_is_zmod_morphism.
HB.instance Definition _ := GRing.isZmodMorphism.Build L L (galTrace U V)
galTrace_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 | galTrace_is_additive | |
galNorm1: galNorm U V 1 = 1.
Proof. by apply: big1 => x _; rewrite rmorph1. 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 | galNorm1 | |
galNormM: {morph galNorm U V : a b / a * b}.
Proof.
by move=> a b /=; rewrite -big_split; apply: eq_bigr => x _; rewrite rmorphM.
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 | galNormM | |
galNormV: {morph galNorm U V : a / a^-1}.
Proof.
by move=> a /=; rewrite -prodfV; apply: eq_bigr => x _; rewrite fmorphV.
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 | galNormV | |
galNormXn : {morph galNorm U V : a / a ^+ n}.
Proof.
move=> a; elim: n => [|n IHn]; first exact: galNorm1.
by rewrite !exprS galNormM IHn.
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 | galNormX | |
galNorm_prod(I : Type) (r : seq I) (P : pred I) (B : I -> L) :
galNorm U V (\prod_(i <- r | P i) B i)
= \prod_(i <- r | P i) galNorm U V (B i).
Proof. exact: (big_morph _ galNormM galNorm1). 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 | galNorm_prod | |
galNorm0: galNorm U V 0 = 0.
Proof. by rewrite /galNorm (bigD1 1%g) ?group1 // rmorph0 /= mul0r. 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 | galNorm0 | |
galNorm_eq0a : (galNorm U V a == 0) = (a == 0).
Proof.
apply/idP/eqP=> [/prodf_eq0[x _] | ->]; last by rewrite galNorm0.
by rewrite fmorph_eq0 => /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 | galNorm_eq0 | |
galTrace_fixedFielda :
a \in E -> galTrace K E a \in fixedField 'Gal(E / K).
Proof.
move=> Ea; apply/fixedFieldP=> [|x galEx].
by apply: rpred_sum => x _; apply: memv_gal.
rewrite {2}/galTrace (reindex_acts 'R _ galEx) ?astabsR //=.
by rewrite rmorph_sum; apply: eq_bigr => y _; rewrite galM ?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 | galTrace_fixedField | |
galTrace_gala x :
a \in E -> x \in 'Gal(E / K) -> galTrace K E (x a) = galTrace K E a.
Proof.
move=> Ea galEx; rewrite {2}/galTrace (reindex_inj (mulgI x)).
by apply: eq_big => [b | b _]; rewrite ?groupMl // galM ?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 | galTrace_gal | |
galNorm_fixedFielda :
a \in E -> galNorm K E a \in fixedField 'Gal(E / K).
Proof.
move=> Ea; apply/fixedFieldP=> [|x galEx].
by apply: rpred_prod => x _; apply: memv_gal.
rewrite {2}/galNorm (reindex_acts 'R _ galEx) ?astabsR //=.
by rewrite rmorph_prod; apply: eq_bigr => y _; rewrite galM ?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 | galNorm_fixedField | |
galNorm_gala x :
a \in E -> x \in 'Gal(E / K) -> galNorm K E (x a) = galNorm K E a.
Proof.
move=> Ea galEx; rewrite {2}/galNorm (reindex_inj (mulgI x)).
by apply: eq_big => [b | b _]; rewrite ?groupMl // galM ?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 | galNorm_gal | |
normalFieldU V := [forall x in kAEndf U, x @: 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 | normalField | |
normalField_kAutK M E f :
(K <= M <= E)%VS -> normalField K M -> kAut K E f -> kAut K M f.
Proof.
case/andP=> sKM sME nKM /kAut_to_gal[x galEx /(sub_in1 (subvP sME))Df].
have sKE := subv_trans sKM sME; rewrite gal_kHom // in galEx.
rewrite (kAut_eq sKM Df) /kAut (kHomSr sME) //= (forall_inP nKM) // inE.
by rewrite kAutfE; apply/kAHomP; apply: (kAHomP galEx).
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 | normalField_kAut | |
normalFieldPK E :
reflect {in E, forall a, exists2 r,
all [in E] r & minPoly K a = \prod_(b <- r) ('X - b%:P)}
(normalField K E).
Proof.
apply: (iffP eqfun_inP) => [nKE a Ea | nKE x]; last first.
rewrite inE kAutfE => homKx; suffices: kAut K E x by case/andP=> _ /eqP.
rewrite kAutE (kHomSr (subvf E)) //=; apply/subvP=> _ /memv_imgP[a Ea ->].
have [r /allP/=srE splitEa] := nKE a Ea.
rewrite srE // -root_prod_XsubC -splitEa.
by rewrite -(kHom_poly_id homKx (minPolyOver K a)) fmorph_root root_minPoly.
have [r /eqP splitKa] := splitting_field_normal K a.
exists r => //; apply/allP => b; rewrite -root_prod_XsubC -splitKa => pKa_b_0.
pose y := kHomExtend K \1 a b; have [hom1K lf1p] := (kHom1 K K, lfun1_poly).
have homKy: kHom K <<K; a>> y by apply/kHomExtendP; rewrite ?lf1p.
have [[g Dy] [idKy _]] := (kHom_to_AEnd homKy, kHomP_tmp homKy).
have <-: g a = b by rewrite -Dy ?memv_adjoin // (kHomExtend_val hom1K) ?lf1p.
suffices /nKE <-: g \in kAEndf K by apply: memv_img.
by rewrite inE kAutfE; apply/kAHomP=> c Kc; rewrite -Dy ?subvP_adjoin ?idKy.
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 | normalFieldP | |
normalFieldfK : normalField K {:L}.
Proof.
apply/normalFieldP=> a _; have [r /eqP->] := splitting_field_normal K a.
by exists r => //; apply/allP=> b; rewrite /= memvf.
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 | normalFieldf | |
normalFieldSK M E : (K <= M)%VS -> normalField K E -> normalField M E.
Proof.
move=> sKM /normalFieldP nKE; apply/normalFieldP=> a Ea.
have [r /allP Er splitKa] := nKE a Ea.
have /dvdp_prod_XsubC[m splitMa]: minPoly M a %| \prod_(b <- r) ('X - b%:P).
by rewrite -splitKa minPolyS.
exists (mask m r); first by apply/allP=> b /mem_mask/Er.
by apply/eqP; rewrite -eqp_monic ?monic_prod_XsubC ?monic_minPoly.
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 | normalFieldS | |
splitting_normalFieldE K :
(K <= E)%VS ->
reflect (exists2 p, p \is a polyOver K & splittingFieldFor K p E)
(normalField K E).
Proof.
move=> sKE; apply: (iffP idP) => [nKE| [p Kp [rs Dp defE]]]; last first.
apply/forall_inP=> g /[!(inE, kAutE)] /andP[homKg _].
rewrite -dimv_leqif_eq ?limg_dim_eq ?(eqP (AEnd_lker0 g)) ?capv0 //.
rewrite -defE aimg_adjoin_seq; have [_ /fixedSpace_limg->] := andP homKg.
apply/adjoin_seqSr=> _ /mapP[a rs_a ->].
rewrite -!root_prod_XsubC -!(eqp_root Dp) in rs_a *.
by apply: kHom_root_id homKg Kp _ rs_a; rewrite ?subvf ?memvf.
pose splitK a r := minPoly K a = \prod_(b <- r) ('X - b%:P).
have{nKE} rK_ a: {r | a \in E -> all [in E] r /\ splitK a r}.
case Ea: (a \in E); last by exists [::].
by have /sig2_eqW[r] := normalFieldP _ _ nKE a Ea; exists r.
have sXE := basis_mem (vbasisP E); set X : seq L := vbasis E in sXE.
exists (\prod_(a <- X) minPoly K a).
by apply: rpred_prod => a _; apply: minPolyOver.
exists (flatten [seq (sval (rK_ a)) | a <- X]).
move/allP: sXE; elim: X => [|a X IHX]; first by rewrite !big_nil eqpxx.
rewrite big_cons /= big_cat /= => /andP[Ea sXE].
by case: (rK_ a) => /= r [] // _ <-; apply/eqp_mull/IHX.
apply/eqP; rewrite eqEsubv; apply/andP; split.
apply/Fadjoin_seqP; split=> // b /flatten_mapP[a /sXE Ea].
by apply/allP; case: rK_ => r /= [].
rewrite -{1}(span_basis (vbasisP E)); apply/span_subvP=> a Xa.
apply/seqv_sub_adjoin/flatten_mapP; exists a => //; rewrite -root_prod_XsubC.
by case: rK_ => /= r [| _ <
... | 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_normalField | |
kHom_to_galK M E f :
(K <= M <= E)%VS -> normalField K E -> kHom K M f ->
{x | x \in 'Gal(E / K) & {in M, f =1 x}}.
Proof.
case/andP=> /subvP sKM /subvP sME nKE KhomMf.
have [[g Df] [idKf _]] := (kHom_to_AEnd KhomMf, kHomP_tmp KhomMf).
suffices /kAut_to_gal[x galEx Dg]: kAut K E g.
by exists x => //= a Ma; rewrite Df // Dg ?sME.
have homKg: kHom K {:L} g by apply/kAHomP=> a Ka; rewrite -Df ?sKM ?idKf.
by rewrite /kAut (kHomSr (subvf _)) // (forall_inP nKE) // inE kAutfE.
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_gal | |
normalField_root_minPolyK E a b :
(K <= E)%VS -> normalField K E -> a \in E -> root (minPoly K a) b ->
exists2 x, x \in 'Gal(E / K) & x a = b.
Proof.
move=> sKE nKE Ea pKa_b_0; pose f := kHomExtend K \1 a b.
have homKa_f: kHom K <<K; a>> f.
by apply: kHomExtendP; rewrite ?kHom1 ?lfun1_poly.
have sK_Ka_E: (K <= <<K; a>> <= E)%VS.
by rewrite subv_adjoin; apply/FadjoinP; rewrite sKE Ea.
have [x galEx Df] := kHom_to_gal sK_Ka_E nKE homKa_f; exists x => //.
by rewrite -Df ?memv_adjoin // (kHomExtend_val (kHom1 K K)) ?lfun1_poly.
Qed.
Arguments normalFieldP {K 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 | normalField_root_minPoly | |
normalField_factorsK E :
(K <= E)%VS ->
reflect {in E, forall a, exists2 r : seq (gal_of E),
r \subset 'Gal(E / K)
& minPoly K a = \prod_(x <- r) ('X - (x a)%:P)}
(normalField K E).
Proof.
move=> sKE; apply: (iffP idP) => [nKE a Ea | nKE]; last first.
apply/normalFieldP=> a Ea; have [r _ ->] := nKE a Ea.
exists [seq x a | x : gal_of E <- r]; last by rewrite big_map.
by rewrite all_map; apply/allP=> b _; apply: memv_gal.
have [r Er splitKa] := normalFieldP nKE a Ea.
pose f b := [pick x in 'Gal(E / K) | x a == b].
exists (pmap f r).
apply/subsetP=> x; rewrite mem_pmap /f => /mapP[b _].
by case: (pickP _) => // c /andP[galEc _] [->].
rewrite splitKa; have{splitKa}: all (root (minPoly K a)) r.
by apply/allP => b; rewrite splitKa root_prod_XsubC.
elim: r Er => /= [|b r IHr]; first by rewrite !big_nil.
case/andP=> Eb Er /andP[pKa_b_0 /(IHr Er){Er}IHr].
have [x galE /eqP xa_b] := normalField_root_minPoly sKE nKE Ea pKa_b_0.
rewrite /(f b); case: (pickP _) => [y /andP[_ /eqP<-]|/(_ x)/andP[]//].
by rewrite !big_cons IHr.
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 | normalField_factors | |
galoisU V := [&& (U <= V)%VS, separable U V & normalField U 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 | galois | |
galoisSK M E : (K <= M <= E)%VS -> galois K E -> galois M E.
Proof.
case/andP=> sKM sME /and3P[_ sepUV nUV].
by rewrite /galois sME (separableSl sKM) ?(normalFieldS sKM).
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 | galoisS | |
galois_dimK E : galois K E -> \dim_K E = #|'Gal(E / K)|.
Proof.
case/and3P=> sKE /eq_adjoin_separable_generator-> // nKE.
set a := separable_generator K E in nKE *.
have [r /allP/=Er splitKa] := normalFieldP nKE a (memv_adjoin K a).
rewrite (dim_sup_field (subv_adjoin K a)) mulnK ?adim_gt0 //.
apply/eqP; rewrite -eqSS -adjoin_degreeE -size_minPoly splitKa size_prod_XsubC.
set n := size r; rewrite eqSS -[n]card_ord.
have x_ (i : 'I_n): {x | x \in 'Gal(<<K; a>> / K) & x a = r`_i}.
apply/sig2_eqW/normalField_root_minPoly; rewrite ?subv_adjoin ?memv_adjoin //.
by rewrite splitKa root_prod_XsubC mem_nth.
have /card_image <-: injective (fun i => s2val (x_ i)).
move=> i j /eqP; case: (x_ i) (x_ j) => y /= galEy Dya [z /= galEx Dza].
rewrite gal_adjoin_eq // Dya Dza nth_uniq // => [/(i =P j)//|].
by rewrite -separable_prod_XsubC -splitKa; apply: separable_generatorP.
apply/eqP/eq_card=> x; apply/codomP/idP=> [[i ->] | galEx]; first by case: x_.
have /(nthP 0) [i ltin Dxa]: x a \in r.
rewrite -root_prod_XsubC -splitKa.
by rewrite root_minPoly_gal ?memv_adjoin ?subv_adjoin.
exists (Ordinal ltin); apply/esym/eqP.
by case: x_ => y /= galEy /eqP; rewrite Dxa gal_adjoin_eq.
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 | galois_dim | |
galois_factorsK E :
(K <= E)%VS ->
reflect {in E, forall a, exists r, let r_a := [seq x a | x : gal_of E <- r] in
[/\ r \subset 'Gal(E / K), uniq r_a
& minPoly K a = \prod_(b <- r_a) ('X - b%:P)]}
(galois K E).
Proof.
move=> sKE; apply: (iffP and3P) => [[_ sepKE nKE] a Ea | galKE].
have [r galEr splitEa] := normalField_factors sKE nKE a Ea.
exists r; rewrite /= -separable_prod_XsubC !big_map -splitEa.
by split=> //; apply: separableP Ea.
split=> //.
apply/separableP => a /galKE[r [_ Ur_a splitKa]].
by rewrite /separable_element splitKa separable_prod_XsubC.
apply/(normalField_factors sKE)=> a /galKE[r [galEr _ ->]].
by rewrite big_map; exists r.
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 | galois_factors | |
splitting_galoisFieldK E :
reflect (exists p, [/\ p \is a polyOver K, separable_poly p
& splittingFieldFor K p E])
(galois K E).
Proof.
apply: (iffP and3P) => [[sKE sepKE nKE]|[p [Kp sep_p [r Dp defE]]]].
rewrite (eq_adjoin_separable_generator sepKE) // in nKE *.
set a := separable_generator K E in nKE *; exists (minPoly K a).
split; first 1 [exact: minPolyOver | exact/separable_generatorP].
have [r /= /allP Er splitKa] := normalFieldP nKE a (memv_adjoin _ _).
exists r; first by rewrite splitKa eqpxx.
apply/eqP; rewrite eqEsubv; apply/andP; split.
by apply/Fadjoin_seqP; split => //; apply: subv_adjoin.
apply/FadjoinP; split; first exact: subv_adjoin_seq.
by rewrite seqv_sub_adjoin // -root_prod_XsubC -splitKa root_minPoly.
have sKE: (K <= E)%VS by rewrite -defE subv_adjoin_seq.
split=> //; last by apply/splitting_normalField=> //; exists p; last exists r.
rewrite -defE; apply/separable_Fadjoin_seq/allP=> a r_a.
by apply/separable_elementP; exists p; rewrite (eqp_root Dp) root_prod_XsubC.
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 | splitting_galoisField | |
galois_fixedFieldK E :
reflect (fixedField 'Gal(E / K) = K) (galois K E).
Proof.
apply: (iffP idP) => [/and3P[sKE /separableP sepKE nKE] | fixedKE].
apply/eqP; rewrite eqEsubv galois_connection_subv ?andbT //.
apply/subvP=> a /mem_fixedFieldP[Ea fixEa]; rewrite -adjoin_deg_eq1.
have [r /allP Er splitKa] := normalFieldP nKE a Ea.
rewrite -eqSS -size_minPoly splitKa size_prod_XsubC eqSS -[1]/(size [:: a]).
have Ur: uniq r by rewrite -separable_prod_XsubC -splitKa; apply: sepKE.
rewrite -uniq_size_uniq {Ur}// => b; rewrite inE -root_prod_XsubC -splitKa.
apply/eqP/idP=> [-> | pKa_b_0]; first exact: root_minPoly.
by have [x /fixEa-> ->] := normalField_root_minPoly sKE nKE Ea pKa_b_0.
have sKE: (K <= E)%VS by rewrite -fixedKE capvSl.
apply/galois_factors=> // a Ea.
pose r_pKa := [seq x a | x : gal_of E in 'Gal(E / K)].
have /fin_all_exists2[x_ galEx_ Dx_a] (b : seq_sub r_pKa) := imageP (valP b).
exists (codom x_); rewrite -map_comp; set r := map _ _.
have r_xa x: x \in 'Gal(E / K) -> x a \in r.
move=> galEx; have r_pKa_xa: x a \in r_pKa by apply/imageP; exists x.
by rewrite [x a](Dx_a (SeqSub r_pKa_xa)); apply: codom_f.
have Ur: uniq r by apply/injectiveP=> b c /=; rewrite -!Dx_a => /val_inj.
split=> //; first by apply/subsetP=> _ /codomP[b ->].
apply/eqP; rewrite -eqp_monic ?monic_minPoly ?monic_prod_XsubC //.
apply/andP; split; last first.
rewrite uniq_roots_dvdp ?uniq_rootsE // all_map.
by apply/allP=> b _ /=; rewrite root_minPoly_gal.
apply: minPoly_dvdp; last by rewri
... | 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 | galois_fixedField | |
mem_galTraceK E a : galois K E -> a \in E -> galTrace K E a \in K.
Proof. by move/galois_fixedField => {2}<- /galTrace_fixedField. 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_galTrace | |
mem_galNormK E a : galois K E -> a \in E -> galNorm K E a \in K.
Proof. by move/galois_fixedField=> {2}<- /galNorm_fixedField. 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_galNorm | |
gal_independent_contraE (P : pred (gal_of E)) (c_ : gal_of E -> L) x :
P x -> c_ x != 0 ->
exists2 a, a \in E & \sum_(y | P y) c_ y * y a != 0.
Proof.
have [n] := ubnP #|P|; elim: n c_ x P => // n IHn c_ x P lePn Px nz_cx.
rewrite ltnS (cardD1x Px) in lePn; move/IHn: lePn => {n IHn}/=IH_P.
have [/eqfun_inP c_Px'_0 | ] := boolP [forall (y | P y && (y != x)), c_ y == 0].
exists 1; rewrite ?mem1v // (bigD1 x Px) /= rmorph1 mulr1.
by rewrite big1 ?addr0 // => y /c_Px'_0->; rewrite mul0r.
case/forall_inPn => y Px'y nz_cy.
have [Py /gal_eqP/eqlfun_inP/subvPn[a Ea]] := andP Px'y.
rewrite memv_ker !lfun_simp => nz_yxa; pose d_ y := c_ y * (y a - x a).
have /IH_P[//|b Eb nz_sumb]: d_ y != 0 by rewrite mulf_neq0.
have [sumb_0|] := eqVneq (\sum_(z | P z) c_ z * z b) 0; last by exists b.
exists (a * b); first exact: rpredM.
rewrite -subr_eq0 -[z in _ - z](mulr0 (x a)) -[in z in _ - z]sumb_0.
rewrite mulr_sumr -sumrB (bigD1 x Px) rmorphM /= mulrCA subrr add0r.
congr (_ != 0): nz_sumb; apply: eq_bigr => z _.
by rewrite mulrCA rmorphM -mulrBr -mulrBl mulrA.
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 | gal_independent_contra | |
gal_independentE (P : pred (gal_of E)) (c_ : gal_of E -> L) :
(forall a, a \in E -> \sum_(x | P x) c_ x * x a = 0) ->
(forall x, P x -> c_ x = 0).
Proof.
move=> sum_cP_0 x Px; apply/eqP/idPn=> /(gal_independent_contra Px)[a Ea].
by rewrite sum_cP_0 ?eqxx.
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 | gal_independent | |
Hilbert's_theorem_90K E x a :
generator 'Gal(E / K) x -> a \in E ->
reflect (exists2 b, b \in E /\ b != 0 & a = b / x b) (galNorm K E a == 1).
Proof.
move/(_ =P <[x]>)=> DgalE Ea.
have galEx: x \in 'Gal(E / K) by rewrite DgalE cycle_id.
apply: (iffP eqP) => [normEa1 | [b [Eb nzb] ->]]; last first.
by rewrite galNormM galNormV galNorm_gal // mulfV // galNorm_eq0.
have [x1 | ntx] := eqVneq x 1%g.
exists 1; first by rewrite mem1v oner_neq0.
by rewrite -{1}normEa1 /galNorm DgalE x1 cycle1 big_set1 !gal_id divr1.
pose c_ y := \prod_(i < invm (injm_Zpm x) y) (x ^+ i)%g a.
have nz_c1: c_ 1%g != 0 by rewrite /c_ morph1 big_ord0 oner_neq0.
have [d] := @gal_independent_contra _ [in 'Gal(E / K)] _ _ (group1 _) nz_c1.
set b := \sum_(y in _) _ => Ed nz_b; exists b.
split=> //; apply: rpred_sum => y galEy.
by apply: rpredM; first apply: rpred_prod => i _; apply: memv_gal.
apply: canRL (mulfK _) _; first by rewrite fmorph_eq0.
rewrite rmorph_sum mulr_sumr [b](reindex_acts 'R _ galEx) ?astabsR //=.
apply: eq_bigr => y galEy; rewrite galM // rmorphM mulrA; congr (_ * _).
have /morphimP[/= i _ _ ->] /=: y \in Zpm @* Zp #[x] by rewrite im_Zpm -DgalE.
have <-: Zpm (i + 1) = (Zpm i * x)%g by rewrite morphM ?mem_Zp ?order_gt1.
rewrite /c_ !invmE ?mem_Zp ?order_gt1 //= addn1; set n := _.+2.
transitivity (\prod_(j < i.+1) (x ^+ j)%g a).
rewrite big_ord_recl gal_id rmorph_prod; congr (_ * _).
by apply: eq_bigr => j _; rewrite expgSr galM ?lfunE.
have [/modn_small->//||->] := ltngtP i.+1 n; first by 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 | Hilbert's_theorem_90 | |
gal_matrix:
{w : #|A|.-tuple L | {subset w <= E} /\ 0 \notin w &
[/\ \matrix_(i, j < #|A|) enum_val i (tnth w j) \in unitmx,
directv (\sum_i K * <[tnth w i]>) &
group_set A -> (\sum_i K * <[tnth w i]>)%VS = E] }.
Proof.
pose nzE (w : #|A|.-tuple L) := {subset w <= E} /\ 0 \notin w.
pose M w := \matrix_(i, j < #|A|) nth 1%g (enum A) i (tnth w j).
have [w [Ew nzw] uM]: {w : #|A|.-tuple L | nzE w & M w \in unitmx}.
rewrite {}/nzE {}/M cardE; have: uniq (enum A) := enum_uniq _.
elim: (enum A) => [|x s IHs] Uxs.
by exists [tuple]; rewrite // flatmx0 -(flatmx0 1%:M) unitmx1.
have [s'x Us]: x \notin s /\ uniq s by apply/andP.
have{IHs} [w [Ew nzw] uM] := IHs Us; set M := \matrix_(i, j) _ in uM.
pose a := \row_i x (tnth w i) *m invmx M.
pose c_ y := oapp (a 0) (-1) (insub (index y s)).
have cx_n1 : c_ x = -1 by rewrite /c_ insubN ?index_mem.
have nz_cx : c_ x != 0 by rewrite cx_n1 oppr_eq0 oner_neq0.
have Px: [pred y in x :: s] x := mem_head x s.
have{Px nz_cx} /sig2W[w0 Ew0 nzS] := gal_independent_contra Px nz_cx.
exists [tuple of cons w0 w].
split; first by apply/allP; rewrite /= Ew0; apply/allP.
rewrite inE negb_or (contraNneq _ nzS) // => <-.
by rewrite big1 // => y _; rewrite rmorph0 mulr0.
rewrite unitmxE -[\det _]mul1r; set M1 := \matrix_(i, j < 1 + size s) _.
have <-: \det (block_mx 1 (- a) 0 1%:M) = 1 by rewrite det_ublock !det1 mulr1.
rewrite -det_mulmx -[M1]submxK mulmx_block !mul0mx !mul1mx !add0r !mulNmx.
have ->:
... | 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 | gal_matrix | |
dim_fixedFieldE (G : {group gal_of E}) : #|G| = \dim_(fixedField G) E.
Proof.
have [w [_ nzw] [_ Edirect /(_ (groupP G))defE]] := gal_matrix G.
set n := #|G|; set m := \dim (fixedField G); rewrite -defE (directvP Edirect).
rewrite -[n]card_ord -(@mulnK #|'I_n| m) ?adim_gt0 //= -sum_nat_const.
congr (_ %/ _)%N; apply: eq_bigr => i _.
by rewrite dim_cosetv ?(memPn nzw) ?mem_tnth.
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 | dim_fixedField | |
dim_fixed_galoisK E (G : {group gal_of E}) :
galois K E -> G \subset 'Gal(E / K) ->
\dim_K (fixedField G) = #|'Gal(E / K) : G|.
Proof.
move=> galE sGgal; have [sFE _ _] := and3P galE; apply/eqP.
rewrite -divgS // eqn_div ?cardSg // dim_fixedField -galois_dim //.
by rewrite mulnC muln_divA ?divnK ?field_dimS ?capvSl -?galois_connection.
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 | dim_fixed_galois | |
gal_fixedFieldE (G : {group gal_of E}): 'Gal(E / fixedField G) = G.
Proof.
apply/esym/eqP; rewrite eqEcard galois_connection_subset /= (dim_fixedField G).
rewrite galois_dim //; apply/galois_fixedField/eqP.
rewrite eqEsubv galois_connection_subv ?capvSl //.
by rewrite fixedFieldS ?galois_connection_subset.
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 | gal_fixedField | |
gal_generatedE (A : {set gal_of E}) : 'Gal(E / fixedField A) = <<A>>.
Proof.
apply/eqP; rewrite eqEsubset gen_subG galois_connection_subset.
by rewrite -[<<A>>]gal_fixedField galS // fixedFieldS // subset_gen.
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 | gal_generated | |
fixedField_galoisE (A : {set gal_of E}): galois (fixedField A) E.
Proof.
have: galois (fixedField <<A>>) E.
by apply/galois_fixedField; rewrite gal_fixedField.
by apply: galoisS; rewrite capvSl fixedFieldS // subset_gen.
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 | fixedField_galois | |
normalField_galois: galois K M.
Proof.
have [[sKM sME] [_ sepKE nKE]] := (andP sKME, and3P galKE).
by rewrite /galois sKM (separableSr sME).
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 | normalField_galois | |
normalField_cast(x : gal_of E) : gal_of M := gal M x. | 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 | normalField_cast | |
normalField_cast_eqx :
x \in 'Gal(E / K) -> {in M, normalField_cast x =1 x}.
Proof.
have [sKM sME] := andP sKME; have sKE := subv_trans sKM sME.
rewrite gal_kAut // => /(normalField_kAut sKME nKM).
by rewrite kAutE => /andP[_ /galK].
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 | normalField_cast_eq | |
normalField_castM:
{in 'Gal(E / K) &, {morph normalField_cast : x y / (x * y)%g}}.
Proof.
move=> x y galEx galEy /=; apply/eqP/gal_eqP => a Ma.
have Ea: a \in E by have [_ /subvP->] := andP sKME.
rewrite normalField_cast_eq ?groupM ?galM //=.
by rewrite normalField_cast_eq ?memv_gal // normalField_cast_eq.
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 | normalField_castM | |
normalField_cast_morphism:= Morphism normalField_castM. | 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 | normalField_cast_morphism | |
normalField_ker: 'ker normalField_cast = 'Gal(E / M).
Proof.
have [sKM sME] := andP sKME.
apply/setP=> x; apply/idP/idP=> [kerMx | galEMx].
rewrite gal_kHom //; apply/kAHomP=> a Ma.
by rewrite -normalField_cast_eq ?(dom_ker kerMx) // (mker kerMx) gal_id.
have galEM: x \in 'Gal(E / K) := subsetP (galS E sKM) x galEMx.
apply/kerP=> //; apply/eqP/gal_eqP=> a Ma.
by rewrite normalField_cast_eq // gal_id (fixed_gal sME).
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 | normalField_ker | |
normalField_normal: 'Gal(E / M) <| 'Gal(E / K).
Proof. by rewrite -normalField_ker ker_normal. 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 | normalField_normal | |
normalField_img: normalField_cast @* 'Gal(E / K) = 'Gal(M / K).
Proof.
have [[sKM sME] [sKE _ nKE]] := (andP sKME, and3P galKE).
apply/setP=> x; apply/idP/idP=> [/morphimP[{}x galEx _ ->] | galMx].
rewrite gal_kHom //; apply/kAHomP=> a Ka; have Ma := subvP sKM a Ka.
by rewrite normalField_cast_eq // (fixed_gal sKE).
have /(kHom_to_gal sKME nKE)[y galEy eq_xy]: kHom K M x by rewrite -gal_kHom.
apply/morphimP; exists y => //; apply/eqP/gal_eqP => a Ha.
by rewrite normalField_cast_eq // eq_xy.
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 | normalField_img | |
normalField_isom:
{f : {morphism ('Gal(E / K) / 'Gal(E / M)) >-> gal_of M} |
isom ('Gal(E / K) / 'Gal (E / M)) 'Gal(M / K) f
& (forall A, f @* (A / 'Gal(E / M)) = normalField_cast @* A)
/\ {in 'Gal(E / K) & M, forall x, f (coset 'Gal (E / M) x) =1 x} }%g.
Proof.
have:= first_isom normalField_cast_morphism; rewrite normalField_ker.
case=> f injf Df; exists f; first by apply/isomP; rewrite Df normalField_img.
split=> [//|x a galEx /normalField_cast_eq<- //]; congr ((_ : gal_of M) a).
apply: set1_inj; rewrite -!morphim_set1 ?mem_quotient ?Df //.
by rewrite (subsetP (normal_norm normalField_normal)).
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 | normalField_isom | |
normalField_isog: 'Gal(E / K) / 'Gal(E / M) \isog 'Gal(M / K).
Proof. by rewrite -normalField_ker -normalField_img first_isog. 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 | normalField_isog | |
normal_fixedField_galois: galois K (fixedField G).
Proof.
have [[sKE sepKE nKE] [sGgal nGgal]] := (and3P galKE, andP nsGgalE).
rewrite /galois -(galois_connection _ sKE) sGgal.
rewrite (separableSr _ sepKE) ?capvSl //; apply/forall_inP=> f autKf.
rewrite eqEdim limg_dim_eq ?(eqP (AEnd_lker0 _)) ?capv0 // leqnn andbT.
apply/subvP => _ /memv_imgP[a /mem_fixedFieldP[Ea cGa] ->].
have /kAut_to_gal[x galEx -> //]: kAut K E f.
rewrite /kAut (forall_inP nKE) // andbT; apply/kAHomP.
by move: autKf; rewrite inE kAutfE => /kHomP_tmp[].
apply/fixedFieldP=> [|y Gy]; first exact: memv_gal.
by rewrite -galM // conjgCV galM //= cGa // memJ_norm ?groupV ?(subsetP nGgal).
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 | normal_fixedField_galois | |
monic_irreducible_poly(p : {poly R}) :=
((irreducible_poly p) * (p \is monic))%type.
Hypothesis hI : monic_irreducible_poly h. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | monic_irreducible_poly | |
qfpoly: monic_irreducible_poly h -> predArgType :=
fun=> {poly %/ h}. | Definition | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qfpoly | |
Definition_ := GRing.NzRing.on {poly %/ h with hI}. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | Definition | |
Definition_ := Finite.on {poly %/ h with hI}. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | Definition | |
Definition_ := GRing.ComUnitRing.on {poly %/ h with hI}. | HB.instance | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | Definition | |
mk_monicE: mk_monic h = h.
Proof. by rewrite /mk_monic !hI. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | mk_monicE | |
coprimep_unit(p : {poly %/ h}) : p != 0%R -> coprimep hQ p.
Proof.
move=> pNZ.
rewrite irreducible_poly_coprime //; last first.
by case: hI; rewrite mk_monicE.
apply: contra pNZ => H; case: eqP => // /eqP /dvdp_leq /(_ H).
by rewrite leqNgt size_mk_monic.
Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | coprimep_unit | |
qpoly_mulVp(p : {poly %/ h}) : p != 0%R -> (qpoly_inv p * p = 1)%R.
Proof. by move=> pNZ; apply/qpoly_mulVz/coprimep_unit. Qed. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qpoly_mulVp | |
qpoly_inv0: qpoly_inv 0%R = 0%R :> {poly %/ h}.
Proof.
rewrite /qpoly_inv /= coprimep0 -size_poly_eq1.
rewrite [in X in X == _]mk_monicE.
by have [[]] := hI; case: size => [|[]].
Qed.
HB.instance Definition _ := GRing.ComUnitRing_isField.Build {poly %/ h with hI}
coprimep_unit.
HB.instance Definition _ := GRing.UnitAlgebra.on {poly %/ h with hI}.
HB.instance Definition _ := Vector.on {poly %/ h with hI}. | Lemma | field | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import fintype tuple div bigop binomial finset finfun",
"From mathcomp Require Import ssralg countalg finalg poly polydiv qpoly perm",
"From mathcomp Require Import f... | field/qfpoly.v | qpoly_inv0 |
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