statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
kAEnd_norm K E : kAEnd K E \subset 'N(kAEndf E)%g. | Proof.
apply/subsetP=> x; rewrite -groupV 2!in_set => /andP[_ /eqP ExE].
apply/subsetP=> _ /imsetP[y homEy ->]; rewrite !in_set !kAutfE in homEy *.
apply/kAHomP=> u Eu; have idEy := kAHomP homEy; rewrite -ExE in idEy.
rewrite !(@lfunE _ _ L) /= (@lfunE _ _ L) /= idEy ?memv_img //.
by rewrite lker0_lfunVK ?AEnd_lker0.
Q... | Lemma | kAEnd_norm | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"apply",
"groupV",
"imsetP",
"in_set",
"kAEnd",
"kAEndf",
"kAHomP",
"kAutfE",
"lfunE",
"lker0_lfunVK",
"memv_img",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_kAut_coset K 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 | mem_kAut_coset | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"coset",
"inE",
"kAEnd_norm",
"kAEndf",
"kAut",
"rcoset_refl",
"subsetP",
"val_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_mem_eqP E (x y : coset_of (kAEndf E)) f g :
f \in x -> g \in y -> reflect {in E, f =1 g} (x == y). | Proof.
move=> x_f y_g; rewrite -(coset_mem x_f) -(coset_mem y_g).
have [Nf Ng] := (subsetP (coset_norm x) f x_f, subsetP (coset_norm y) g y_g).
rewrite (sameP eqP (rcoset_kercosetP Nf Ng)) mem_rcoset inE kAutfE.
apply: (iffP kAHomP) => idEfg u Eu.
by rewrite -(mulgKV g f) lfunE /= idEfg.
by rewrite (@lfunE _ _ L) /= ... | Lemma | aut_mem_eqP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"apply",
"coset_mem",
"coset_norm",
"coset_of",
"inE",
"kAEndf",
"kAHomP",
"kAutfE",
"lfunE",
"lker0_lfunK",
"mem_rcoset",
"mulgKV",
"rcoset_kercosetP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_of | := Gal of [subg kAEnd_group 1 <<V>> / kAEndf (agenv V)]. | Inductive | gal_of | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"agenv",
"kAEnd_group",
"kAEndf",
"subg"
] | the argument of [subg _] is syntactically a group. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gal (f : 'AEnd(L)) | := Gal (subg _ (coset _ f)). | Definition | gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"coset",
"subg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_sgval x | := let: Gal u := x in u. | Definition | gal_sgval | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_sgvalK : cancel gal_sgval Gal. | Proof. by case. Qed. | Fact | gal_sgvalK | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_sgval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_sgval_inj | := can_inj gal_sgvalK. | Let | gal_sgval_inj | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_sgvalK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_one | := Gal 1%g. | Definition | gal_one | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_inv x | := Gal (gal_sgval x)^-1. | Definition | gal_inv | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_sgval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_mul x y | := Gal (gal_sgval x * gal_sgval y). | Definition | gal_mul | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_sgval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_oneP : left_id gal_one gal_mul. | Proof. by move=> x; apply/gal_sgval_inj/mul1g. Qed. | Fact | gal_oneP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"gal_mul",
"gal_one",
"gal_sgval_inj",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_invP : left_inverse gal_one gal_inv gal_mul. | Proof. by move=> x; apply/gal_sgval_inj/mulVg. Qed. | Fact | gal_invP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"gal_inv",
"gal_mul",
"gal_one",
"gal_sgval_inj",
"mulVg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_mulP : associative gal_mul. | Proof. by move=> x y z; apply/gal_sgval_inj/mulgA. Qed. | Fact | gal_mulP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"gal_mul",
"gal_sgval_inj",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_repr u : 'AEnd(L) | := repr (sgval (gal_sgval u)). | Coercion | gal_repr | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_sgval",
"repr",
"sgval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Fact | gal_is_morphism | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"agenv",
"gal",
"kAEnd",
"kAEnd_norm",
"mem_morphim",
"morphM",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_morphism | := Morphism gal_is_morphism. | Canonical | gal_morphism | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_is_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_reprK : cancel gal_repr gal. | Proof. by case=> x; rewrite /gal coset_reprK sgvalK. Qed. | Lemma | gal_reprK | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"coset_reprK",
"gal",
"gal_repr",
"sgvalK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_repr_inj : injective gal_repr. | Proof. exact: can_inj gal_reprK. Qed. | Lemma | gal_repr_inj | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_repr",
"gal_reprK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_AEnd x : 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 | gal_AEnd | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"agenv",
"fixedSpace_limg",
"gal_repr",
"gal_sgval",
"groupMr",
"inE",
"kAEnd",
"kAut1E",
"kAutE",
"morphimP",
"repr_rcosetP",
"val_coset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_eqP E {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 | gal_eqP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"aut_mem_eqP",
"gal_of",
"mem_repr_coset",
"subfield_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galK E (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 | galK | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"aut_mem_eqP",
"eqxx",
"gal",
"inE",
"kAut1E",
"mem_kAut_coset",
"mem_quotient",
"mem_repr_coset",
"subfield_closed",
"subgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_galP E (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 | eq_galP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"gal",
"galK",
"gal_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
limg_gal E (x : gal_of E) : (x @: E)%VS = E. | Proof. by have:= gal_AEnd x; rewrite inE subfield_closed => /andP[_ /eqP]. Qed. | Lemma | limg_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_AEnd",
"gal_of",
"inE",
"subfield_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memv_gal E (x : gal_of E) a : a \in E -> x a \in E. | Proof. by move/(memv_img x); rewrite limg_gal. Qed. | Lemma | memv_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_of",
"limg_gal",
"memv_img"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_id E a : (1 : gal_of E)%g a = a. | Proof. by rewrite /gal_repr repr_coset1 id_lfunE. Qed. | Lemma | gal_id | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_of",
"gal_repr",
"id_lfunE",
"repr_coset1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galM E (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 | galM | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"comp_lfunE",
"eq_galP",
"gal_AEnd",
"gal_of",
"gal_reprK",
"limg_comp",
"limg_gal",
"morphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galV E (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 | galV | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"apply",
"galM",
"gal_id",
"gal_of",
"lker0_lfunK",
"mulVg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galoisG V U | := gal V @* <<kAEnd (U :&: V) V>>. | Definition | galoisG | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal",
"kAEnd"
] | Standard mathematical notation for 'Gal(E / K) puts the larger field first. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"''Gal' ( V / U )" | := (galoisG V U) : group_scope. | Notation | ''Gal' ( V / U ) | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galoisG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galoisG_group E U | := Eval hnf in [group of (galoisG E U)]. | Canonical | galoisG_group | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galoisG",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Gal' ( V / U )" | := (galoisG_group V U) : Group_scope. | Notation | ''Gal' ( V / U ) | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galoisG_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_cap U V : 'Gal(V / U) = 'Gal(V / U :&: V). | Proof. by rewrite /galoisG -capvA capvv. Qed. | Lemma | gal_cap | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"capvA",
"capvv",
"galoisG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_kAut K 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 | gal_kAut | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capv_idPl",
"galK",
"gal_AEnd",
"gal_reprK",
"genGid",
"inE",
"kAut",
"kAut1E",
"kAut_eq",
"mem_gen",
"morphimP",
"sKE",
"subfield_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_kHom K 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 | gal_kHom | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"eqxx",
"gal_kAut",
"kAut",
"kHom",
"limg_gal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAut_to_gal K 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 // -kAut1... | Lemma | kAut_to_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"capvSl",
"capvSr",
"eq_in_limg",
"gal",
"galK",
"gal_of",
"genGid",
"inE",
"kAut",
"kAut1E",
"kAutS",
"kHomSl",
"kHom_eq",
"kHom_to_AEnd",
"last",
"mem_morphim",
"sub1v",
"subfield_closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixed_gal K 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 | fixed_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal_kHom",
"kAHomP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedPoly_gal K 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 | fixedPoly_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"coef_map",
"fixed_gal",
"map_poly",
"polyOver",
"polyOverP",
"polyP",
"sKE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_minPoly_gal K 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 | root_minPoly_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"fixedPoly_gal",
"gal_kHom",
"kHom",
"kHom_root",
"minPoly",
"minPolyOver",
"polyOverS",
"root",
"root_minPoly",
"sKE",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_adjoin_eq K 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 | gal_adjoin_eq | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"Fadjoin_polyP",
"apply",
"fixedPoly_gal",
"gal_eqP",
"horner_map",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galS K 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 | galS | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capvS",
"capvSr",
"gal_cap",
"gal_kAut",
"kAutS",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_conjg K 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].
... | Lemma | gal_conjg | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"apply",
"capvSr",
"conjgC",
"eqEsubset",
"eqEsubv",
"eq_in_limg",
"fixed_gal",
"galM",
"gal_cap",
"gal_id",
"gal_kHom",
"imsetP",
"kAHomP",
"last",
"lfunE",
"lim1g",
"limg_cap",
"limg_comp",
"limg_gal",
"lker0P",
"memv_cap",
"memv_capP",
"memv_gal",
"me... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedField V (A : {set gal_of V}) | :=
(V :&: \bigcap_(x in A) fixedSpace x)%VS. | Definition | fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"fixedSpace",
"gal_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedFieldP E {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 | fixedFieldP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fixedField",
"fixedSpaceP",
"gal_of",
"memv_cap",
"subv_bigcapP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_fixedFieldP E (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. | Lemma | mem_fixedFieldP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"fixedField",
"fixedFieldP",
"gal_of",
"memv_capP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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. | Fact | fixedField_is_aspace | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"big_rec",
"capvA",
"fixedField",
"gal_of",
"is_aspace",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedField_aspace E A : {subfield L} | :=
ASpace (@fixedField_is_aspace E A). | Canonical | fixedField_aspace | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"fixedField_is_aspace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedField_bound E (A : {set gal_of E}) : (fixedField A <= E)%VS. | Proof. exact: capvSl. Qed. | Lemma | fixedField_bound | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"capvSl",
"fixedField",
"gal_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedFieldS E (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 | fixedFieldS | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fixedField",
"fixedFieldP",
"gal_of",
"last",
"mem_fixedFieldP",
"subsetP",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_connection_subv K 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 | galois_connection_subv | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fixedField",
"fixedFieldP",
"fixed_gal",
"sKE",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_connection_subset E (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 | galois_connection_subset | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capvSl",
"fixedField",
"gal_kAut",
"gal_of",
"kAHomP",
"kAutE",
"limg_gal",
"mem_fixedFieldP",
"subsetP",
"subvv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_connection K 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 | galois_connection | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fixedField",
"fixedFieldS",
"galS",
"gal_of",
"galois_connection_subset",
"galois_connection_subv",
"sKE",
"subset_trans",
"subv_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galTrace U V a | := \sum_(x in 'Gal(V / U)) (x a). | Definition | galTrace | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm U V a | := \prod_(x in 'Gal(V / U)) (x a). | Definition | galNorm | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galTrace_is_zmod_morphism : zmod_morphism (galTrace U V). | Proof.
by move=> a b /=; rewrite -sumrB; apply: eq_bigr => x _; rewrite rmorphB.
Qed. | Fact | galTrace_is_zmod_morphism | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"eq_bigr",
"galTrace",
"rmorphB",
"sumrB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galTrace_is_additive | := galTrace_is_zmod_morphism. | Definition | galTrace_is_additive | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galTrace_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm1 : galNorm U V 1 = 1. | Proof. by apply: big1 => x _; rewrite rmorph1. Qed. | Lemma | galNorm1 | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"big1",
"galNorm",
"rmorph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | galNormM | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"big_split",
"eq_bigr",
"galNorm",
"rmorphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNormV : {morph galNorm U V : a / a^-1}. | Proof.
by move=> a /=; rewrite -prodfV; apply: eq_bigr => x _; rewrite fmorphV.
Qed. | Lemma | galNormV | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"eq_bigr",
"fmorphV",
"galNorm",
"prodfV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNormX n : {morph galNorm U V : a / a ^+ n}. | Proof.
move=> a; elim: n => [|n IHn]; first exact: galNorm1.
by rewrite !exprS galNormM IHn.
Qed. | Lemma | galNormX | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"exprS",
"galNorm",
"galNorm1",
"galNormM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | galNorm_prod | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"big_morph",
"galNorm",
"galNorm1",
"galNormM",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm0 : galNorm U V 0 = 0. | Proof. by rewrite /galNorm (bigD1 1%g) ?group1 // rmorph0 /= mul0r. Qed. | Lemma | galNorm0 | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"bigD1",
"galNorm",
"group1",
"mul0r",
"rmorph0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm_eq0 a : (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 | galNorm_eq0 | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"fmorph_eq0",
"galNorm",
"galNorm0",
"last",
"prodf_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galTrace_fixedField a :
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 | galTrace_fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"astabsR",
"eq_bigr",
"fixedField",
"fixedFieldP",
"galM",
"galTrace",
"lfunE",
"memv_gal",
"reindex_acts",
"rmorph_sum",
"rpred_sum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galTrace_gal a 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 | galTrace_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"eq_big",
"galM",
"galTrace",
"groupMl",
"lfunE",
"mulgI",
"reindex_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm_fixedField a :
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 | galNorm_fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"astabsR",
"eq_bigr",
"fixedField",
"fixedFieldP",
"galM",
"galNorm",
"lfunE",
"memv_gal",
"reindex_acts",
"rmorph_prod",
"rpred_prod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galNorm_gal a 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 | galNorm_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"eq_big",
"galM",
"galNorm",
"groupMl",
"lfunE",
"mulgI",
"reindex_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField U V | := [forall x in kAEndf U, x @: V == V]%VS. | Definition | normalField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"kAEndf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_kAut K 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 | normalField_kAut | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"forall_inP",
"gal_kHom",
"inE",
"kAHomP",
"kAut",
"kAut_eq",
"kAut_to_gal",
"kAutfE",
"kHomSr",
"normalField",
"sKE",
"subvP",
"subv_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalFieldP K 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 -(... | Lemma | normalFieldP | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"all",
"allP",
"apply",
"eqfun_inP",
"fmorph_root",
"inE",
"kAEndf",
"kAHomP",
"kAut",
"kAutE",
"kAutfE",
"kHom",
"kHom1",
"kHomExtend",
"kHomExtendP",
"kHomExtend_val",
"kHomP_tmp",
"kHomSr",
"kHom_poly_id",
"kHom_to_AEnd",
"last",
"lfun1_poly",
"memv_adjoin",
"memv_im... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalFieldf K : 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 | normalFieldf | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"allP",
"apply",
"memvf",
"normalField",
"normalFieldP",
"splitting_field_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalFieldS K 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 ... | Lemma | normalFieldS | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"allP",
"apply",
"dvdp_prod_XsubC",
"eqp_monic",
"mask",
"mem_mask",
"minPoly",
"minPolyS",
"monic_minPoly",
"monic_prod_XsubC",
"normalField",
"normalFieldP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitting_normalField E 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... | Lemma | splitting_normalField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"AEnd_lker0",
"Fadjoin_seqP",
"adjoin_seqSr",
"aimg_adjoin_seq",
"all",
"allP",
"apply",
"basis_mem",
"big_cat",
"big_cons",
"big_nil",
"capv0",
"dimv_leqif_eq",
"eqEsubv",
"eqp_mull",
"eqp_root",
"eqpxx",
"fixedSpace_limg",
"flatten",
"flatten_mapP",
"forall_inP",
"inE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_to_gal K 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 (sub... | Lemma | kHom_to_gal | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"forall_inP",
"inE",
"kAHomP",
"kAut",
"kAut_to_gal",
"kAutfE",
"kHom",
"kHomP_tmp",
"kHomSr",
"kHom_to_AEnd",
"normalField",
"subvP",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_root_minPoly K 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... | Lemma | normalField_root_minPoly | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"FadjoinP",
"apply",
"kHom",
"kHom1",
"kHomExtend",
"kHomExtendP",
"kHomExtend_val",
"kHom_to_gal",
"lfun1_poly",
"memv_adjoin",
"minPoly",
"normalField",
"root",
"sKE",
"subv_adjoin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_factors K 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 'Ga... | Lemma | normalField_factors | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"all",
"allP",
"all_map",
"apply",
"big_cons",
"big_map",
"big_nil",
"gal_of",
"last",
"mapP",
"mem_pmap",
"memv_gal",
"minPoly",
"normalField",
"normalFieldP",
"normalField_root_minPoly",
"pick",
"pickP",
"pmap",
"root",
"root_prod_XsubC",
"sKE",
"seq",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois U V | := [&& (U <= V)%VS, separable U V & normalField U V]. | Definition | galois | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"normalField",
"separable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galoisS K 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 | galoisS | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galois",
"normalFieldS",
"separableSl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_dim K 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.
se... | Lemma | galois_dim | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"adim_gt0",
"adjoin_degreeE",
"allP",
"apply",
"card_image",
"card_ord",
"codomP",
"dim_sup_field",
"eqSS",
"eq_adjoin_separable_generator",
"eq_card",
"gal_adjoin_eq",
"galois",
"mem_nth",
"memv_adjoin",
"mulnK",
"normalFieldP",
"normalField_root_minPoly",
"nthP",
"nth_uniq",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_factors K 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 /separ... | Lemma | galois_factors | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"big_map",
"galKE",
"gal_of",
"galois",
"minPoly",
"normalField_factors",
"sKE",
"separableP",
"separable_element",
"separable_prod_XsubC",
"seq",
"split",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitting_galoisField K 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 n... | Lemma | splitting_galoisField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"FadjoinP",
"Fadjoin_seqP",
"allP",
"apply",
"eqEsubv",
"eq_adjoin_separable_generator",
"eqp_root",
"eqpxx",
"galois",
"last",
"memv_adjoin",
"minPoly",
"minPolyOver",
"normalFieldP",
"polyOver",
"root_minPoly",
"root_prod_XsubC",
"sKE",
"separable_Fadjoin_seq",
"separable_ele... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galois_fixedField K 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 eqS... | Lemma | galois_fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"adjoin_deg_eq1",
"allP",
"all_map",
"apply",
"big_map",
"capvSl",
"codom",
"codomP",
"codom_f",
"coef_map",
"eqEsubv",
"eqSS",
"eq_bigr",
"eqp_monic",
"fin_all_exists2",
"fixedField",
"fixedFieldP",
"fmorph_inj",
"galM",
"gal_id",
"gal_of",
"galois",
"galois_connection_s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_galTrace K E a : galois K E -> a \in E -> galTrace K E a \in K. | Proof. by move/galois_fixedField => {2}<- /galTrace_fixedField. Qed. | Lemma | mem_galTrace | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galTrace",
"galTrace_fixedField",
"galois",
"galois_fixedField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_galNorm K E a : galois K E -> a \in E -> galNorm K E a \in K. | Proof. by move/galois_fixedField=> {2}<- /galNorm_fixedField. Qed. | Lemma | mem_galNorm | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galNorm",
"galNorm_fixedField",
"galois",
"galois_fixedField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_independent_contra E (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 // => ... | Lemma | gal_independent_contra | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"Px",
"add0r",
"addr0",
"apply",
"big1",
"bigD1",
"cardD1x",
"eqVneq",
"eq_bigr",
"eqfun_inP",
"eqlfun_inP",
"forall_inPn",
"gal_eqP",
"gal_of",
"last",
"lfun_simp",
"ltnS",
"mem1v",
"memv_ker",
"mul0r",
"mulf_neq0",
"mulr0",
"mulr1",
"mulrA",
"mulrBl",
"mulrBr",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_independent E (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 | gal_independent | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"Px",
"apply",
"eqxx",
"gal_independent_contra",
"gal_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hilbert's_theorem_90 K 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}... | Lemma | Hilbert's_theorem_90 | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"Zp",
"Zpm",
"addn1",
"apply",
"astabsR",
"big_imset",
"big_ord0",
"big_ord_recl",
"big_set1",
"cycle1",
"cycle_id",
"divr1",
"eqVneq",
"eq_bigl",
"eq_bigr",
"expgSr",
"fmorph_eq0",
"galM",
"galNorm",
"galNormM",
"galNormV",
"galNorm_eq0",
"galNorm_gal",
"gal_id",
"ga... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
K | := fixedField A. | Let | K | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"fixedField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 [t... | Lemma | gal_matrix | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"Px",
"add0r",
"addrC",
"allP",
"apply",
"asubv",
"big1",
"big_cons",
"big_mkord",
"big_nth",
"big_uniq",
"block_mx",
"capvSl",
"cardE",
"colP",
"contraNneq",
"det1",
"det_lblock",
"det_mulmx",
"det_scalar",
"det_ublock",
"directv",
"directv_sum_independent",
"drsubmx",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_fixedField E (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 | dim_fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"adim_gt0",
"apply",
"card_ord",
"dim",
"dim_cosetv",
"directvP",
"eq_bigr",
"fixedField",
"gal_matrix",
"gal_of",
"group",
"groupP",
"memPn",
"mem_tnth",
"mulnK",
"sum_nat_const"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_fixed_galois K 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 | dim_fixed_galois | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capvSl",
"cardSg",
"dim_fixedField",
"divgS",
"divnK",
"eqn_div",
"field_dimS",
"fixedField",
"gal_of",
"galois",
"galois_connection",
"galois_dim",
"group",
"mulnC",
"muln_divA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_fixedField E (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 | gal_fixedField | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capvSl",
"dim_fixedField",
"eqEcard",
"eqEsubv",
"fixedField",
"fixedFieldS",
"gal_of",
"galois_connection_subset",
"galois_connection_subv",
"galois_dim",
"galois_fixedField",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gal_generated E (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 | gal_generated | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"eqEsubset",
"fixedField",
"fixedFieldS",
"galS",
"gal_fixedField",
"gal_of",
"galois_connection_subset",
"gen_subG",
"subset_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fixedField_galois E (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 | fixedField_galois | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"capvSl",
"fixedField",
"fixedFieldS",
"gal_fixedField",
"gal_of",
"galois",
"galoisS",
"galois_fixedField",
"subset_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galKE : galois K E. | Hypothesis | galKE | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galois"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
(sKME : (K <= M <= E)%VS) (nKM : normalField K M). | Hypothesis | sKME | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"normalField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
normalField_galois : galois K M. | Proof.
have [[sKM sME] [_ sepKE nKE]] := (andP sKME, and3P galKE).
by rewrite /galois sKM (separableSr sME).
Qed. | Lemma | normalField_galois | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galKE",
"galois",
"sKME",
"separableSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_cast (x : gal_of E) : gal_of M | := gal M x. | Definition | normalField_cast | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"gal",
"gal_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalField_cast_eq x :
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 | normalField_cast_eq | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"galK",
"gal_kAut",
"kAutE",
"normalField_cast",
"normalField_kAut",
"sKE",
"sKME",
"subv_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | normalField_castM | field | field/galois.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"ssralg",
"poly",
"polydiv",
"finset",
"fingroup",
"morphism",
"quotient",
"perm",
"action",
"zmodp",
"cyclic",
"ma... | [
"apply",
"galM",
"gal_eqP",
"groupM",
"memv_gal",
"normalField_cast",
"normalField_cast_eq",
"sKME",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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