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 |
|---|---|---|---|---|---|---|---|---|---|---|
primeChar_scaleAr | := (pprimeChar_scaleAr) (only parsing). | Notation | primeChar_scaleAr | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_scaleAr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_abelem | := (pprimeChar_abelem) (only parsing). | Notation | primeChar_abelem | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_pgroup | := (pprimeChar_pgroup) (only parsing). | Notation | primeChar_pgroup | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_pgroup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_primeChar | := (order_pprimeChar) (only parsing). | Notation | order_primeChar | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"order_pprimeChar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_primeChar | := (card_pprimeChar) (only parsing). | Notation | card_primeChar | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"card_pprimeChar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_vectAxiom | := (pprimeChar_vectAxiom) (only parsing). | Notation | primeChar_vectAxiom | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_vectAxiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
primeChar_dimf | := (pprimeChar_dimf) (only parsing). | Notation | primeChar_dimf | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pprimeChar_dimf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order (L : vectType F) (K : {vspace L}) | := (#|F| ^ \dim K)%N. | Let | order | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"dim"
] | do not want to impose the FinVector instance here. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
galL K : galois K {:L}. | Proof.
without loss {K} ->: K / K = 1%AS.
by move=> IH_K; apply: galoisS (IH_K _ (erefl _)); rewrite sub1v subvf.
apply/splitting_galoisField; pose finL := FinFieldExtType L.
exists ('X^#|finL| - 'X); split; first by rewrite rpredB 1?rpredX ?polyOverX.
rewrite (finField_genPoly finL) -big_enum /=.
by rewrite sepa... | Let | galL | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinFieldExtType",
"apply",
"big_enum",
"enum",
"enum_uniq",
"eqpxx",
"finField_genPoly",
"galois",
"galoisS",
"mem_enum",
"memvf",
"polyOverX",
"rpredB",
"rpredX",
"separable_prod_XsubC",
"seqv_sub_adjoin",
"split",
"splitting_galoisField",
"sub1v",
"subvf",
"vspaceP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
galLgen K :
{alpha | generator 'Gal({:L} / K) alpha & forall x, alpha x = x ^+ order K}. | Proof.
without loss{K} ->: K / K = 1%AS; last rewrite /order dimv1 expn1.
case/(_ 1%AS)=> // alpha /eqP-defGalL; rewrite /order dimv1 expn1 => Dalpha.
exists (alpha ^+ \dim K)%g => [|x]; last first.
elim: (\dim K) => [|n IHn]; first by rewrite gal_id.
by rewrite expgSr galM ?memvf // IHn Dalpha expnSr exprM... | Fact | galLgen | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"Build",
"alpha",
"apply",
"big_image",
"card_pprimeChar",
"cycleX",
"cycle_cyclic",
"dim",
"dimv1",
"divn1",
"eq_bigr",
"eq_subG_cyclic",
"expf_card",
"expgSr",
"expn1",
"expnSr",
"expr1n",
"exprM",
"exprMn",
"fA",
"fM",
"field_dimS",
"finField_genPoly",
"finPcharP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finField_galois K E : (K <= E)%VS -> galois K E. | Proof.
move=> sKE; have /galois_fixedField <- := galL E.
rewrite normal_fixedField_galois // -sub_abelian_normal ?galS //.
apply: abelianS (galS _ (sub1v _)) _.
by have [alpha /('Gal(_ / _) =P _)-> _] := galLgen 1; apply: cycle_abelian.
Qed. | Lemma | finField_galois | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"abelianS",
"alpha",
"apply",
"cycle_abelian",
"galL",
"galLgen",
"galS",
"galois",
"galois_fixedField",
"normal_fixedField_galois",
"sKE",
"sub1v",
"sub_abelian_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finField_galois_generator K E :
(K <= E)%VS ->
{alpha | generator 'Gal(E / K) alpha
& {in E, forall x, alpha x = x ^+ order K}}. | Proof.
move=> sKE; have [alpha defGalLK Dalpha] := galLgen K.
have inKL_E: (K <= E <= {:L})%VS by rewrite sKE subvf.
have nKE: normalField K E by have/and3P[] := finField_galois sKE.
have galLKalpha: alpha \in 'Gal({:L} / K).
by rewrite (('Gal(_ / _) =P _) defGalLK) cycle_id.
exists (normalField_cast _ alpha) => [|x ... | Lemma | finField_galois_generator | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"alpha",
"cycle_id",
"finField_galois",
"galLgen",
"generator",
"last",
"morphim_cycle",
"normalField",
"normalField_cast",
"normalField_cast_eq",
"normalField_cast_morphism",
"normalField_img",
"order",
"sKE",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Fermat's_little_theorem (L : fieldExtType F) (K : {subfield L}) a :
(a \in K) = (a ^+ order K == a). | Proof.
move: K a; wlog [{}L -> K a]: L / exists galL : splittingFieldType F, L = galL.
by pose galL := FinSplittingFieldType F L => /(_ galL); apply; exists galL.
have /galois_fixedField fixLK := finField_galois (subvf K).
have [alpha defGalLK Dalpha] := finField_galois_generator (subvf K).
rewrite -Dalpha ?memvf // ... | Lemma | Fermat's_little_theorem | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinSplittingFieldType",
"alpha",
"apply",
"cycle",
"finField_galois",
"finField_galois_generator",
"fixedFieldP",
"fixedField_galois",
"galL",
"gal_generated",
"galois_fixedField",
"memvf",
"order",
"set11",
"set1P",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_poly_extField (F : fieldType) (L : fieldExtType F) | :=
map_poly (in_alg L) : {poly F} -> {poly L}. | Let | map_poly_extField | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"in_alg",
"map_poly",
"poly"
] | The apparently redundant type annotation reduces checking time by 30%. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"p ^%:A" | := (map_poly_extField _ p)
(format "p ^%:A") : ring_scope. | Notation | p ^%:A | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"map_poly_extField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinSplittingFieldFor (F : finFieldType) (p : {poly F}) :
p != 0 -> {L : splittingFieldType F | splittingFieldFor 1 p^%:A {:L}}. | Proof.
have mapXsubC (f : {rmorphism _ -> _}) x:
map_poly f ('X - x%:P) = 'X - (f x)%:P.
by rewrite rmorphB /= map_polyX map_polyC.
move=> nz_p; pose splits q := {zs | q %= \prod_(z <- zs) ('X - z%:P)}.
suffices [L splitLp]: {L : fieldExtType F | splittingFieldFor 1 p^%:A {:L}}.
by exists (FinSplittingFieldType... | Lemma | FinSplittingFieldFor | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinFieldExtType",
"FinSplittingFieldType",
"addn2",
"aimg1",
"aimg_adjoin_seq",
"apply",
"arg_minnP",
"baseFieldType",
"baseField_scaleE",
"big_cons",
"big_map",
"big_nil",
"divpK",
"dvdpN0",
"dvdp_leq",
"dvdp_size_eqp",
"dvdp_trans",
"eq_bigr",
"eq_map_poly",
"eq_sym",
"eqn... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pPrimePowerField p k (m := (p ^ k)%N) :
prime p -> 0 < k -> {Fm : finFieldType | p \in [pchar Fm] & #|Fm| = m}. | Proof.
move=> pr_p k_gt0; have m_gt1: m > 1 by rewrite (ltn_exp2l 0) ?prime_gt1.
have m_gt0 := ltnW m_gt1; have m1_gt0: m.-1 > 0 by rewrite -ltnS prednK.
pose q := 'X^m - 'X; have Dq R: q R = ('X^(m.-1) - 1) * ('X - 0).
by rewrite subr0 mulrBl mul1r -exprSr prednK.
have /FinSplittingFieldFor[/= L splitLq]: q 'F_p != ... | Lemma | pPrimePowerField | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"Da",
"FinFieldExtType",
"FinSplittingFieldFor",
"agenvS",
"apply",
"card_Fp",
"card_uniqP",
"cyclotomic",
"dimv1",
"eqEsubv",
"eq_card",
"eq_sym",
"eqp_root",
"eqp_separable",
"eqp_size",
"expgSr",
"expn1",
"expnSr",
"expr0n",
"exprM",
"exprSr",
"finField_galois_generator"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PrimePowerField | := (pPrimePowerField) (only parsing). | Notation | PrimePowerField | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"pPrimePowerField"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
domR : GRing.integral_domain_axiom R. | Hypothesis | domR | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"integral_domain_axiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
lregR x : x != 0 -> GRing.lreg x. | Proof. by move=> xnz; apply: mulrI0_lreg => y /domR/orP[/idPn | /eqP]. Qed. | Let | lregR | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"domR",
"lreg",
"mulrI0_lreg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finDomain_field : GRing.field_axiom R. | Proof.
move=> x /lregR-regx; apply/unitrP; exists (invF regx 1).
by split; first apply: (regx); rewrite ?mulrA f_invF // mulr1 mul1r.
Qed. | Lemma | finDomain_field | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"apply",
"f_invF",
"field_axiom",
"invF",
"lregR",
"mul1r",
"mulr1",
"mulrA",
"split",
"unitrP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finDomain_mulrC : @commutative R R *%R. | Proof.
have fieldR := finDomain_field.
have [p p_pr pcharRp]: exists2 p, prime p & p \in [pchar R].
have [e /prod_prime_decomp->]: {e | (e > 0)%N & e%:R == 0 :> R}.
by exists #|[set: R]%G|; rewrite // -order_dvdn order_dvdG ?inE.
rewrite big_seq; elim/big_rec: _ => [|[p m] /= n]; first by rewrite oner_eq0.
ca... | Theorem | finDomain_mulrC | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"C_prim_root_exists",
"Cintr_Cyclotomic",
"Sub",
"addKr",
"addrC",
"adim_gt0",
"algC",
"apply",
"bigD1",
"big_filter",
"big_ind",
"big_mkcond",
"big_mkord",
"big_rec",
"big_seq",
"cardD1",
"card_Fp",
"card_preim",
"card_vspace",
"cent1E",
"cent1vP",
"centP",
"center_class... | This is Witt's proof of Wedderburn's little theorem. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
FinDomainFieldType : finFieldType | :=
let cC := GRing.SemiRing_hasCommutativeMul.Build R finDomain_mulrC in
let cR : comUnitRingType := HB.pack R cC in
let iC := GRing.ComUnitRing_isIntegral.Build cR domR in
let iR : finIdomainType := HB.pack cR iC in
let fC := GRing.UnitRing_isField.Build iR finDomain_field in
HB.pack iR fC. | Definition | FinDomainFieldType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"Build",
"domR",
"finDomain_field",
"finDomain_mulrC",
"iC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinDomainSplittingFieldType_pchar p (pcharRp : p \in [pchar R]) | :=
SplittingField.clone 'F_p R (@pPrimeCharType p FinDomainFieldType pcharRp). | Definition | FinDomainSplittingFieldType_pchar | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinDomainFieldType",
"clone",
"pPrimeCharType",
"pchar",
"pcharRp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinDomainSplittingFieldType | := (FinDomainSplittingFieldType_pchar) (only parsing). | Notation | FinDomainSplittingFieldType | field | field/finfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"div",
"tuple",
"bigop",
"prime",
"finset",
"fingroup",
"ssralg",
"poly",
"polydiv",
"morphism",
"action",
"countalg",
"finalg",
"zmodp",
"cyclic",
"c... | [
"FinDomainSplittingFieldType_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splittingFieldFor (U : {vspace L}) (p : {poly L}) (V : {vspace L}) | :=
exists2 rs, p %= \prod_(z <- rs) ('X - z%:P) & <<U & rs>>%VS = V. | Definition | splittingFieldFor | 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... | [
"poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splittingFieldForS (K M E : {subfield L}) p :
(K <= M)%VS -> (M <= E)%VS ->
splittingFieldFor K p E -> splittingFieldFor M p E. | Proof.
move=> sKM sKE [rs Dp genL]; exists rs => //; apply/eqP.
rewrite eqEsubv -[in X in _ && (X <= _)%VS]genL adjoin_seqSl // andbT.
by apply/Fadjoin_seqP; split; rewrite // -genL; apply: seqv_sub_adjoin.
Qed. | Lemma | splittingFieldForS | 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_seqP",
"adjoin_seqSl",
"apply",
"eqEsubv",
"sKE",
"seqv_sub_adjoin",
"split",
"splittingFieldFor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom U V f | := ahom_in V f && (U <= fixedSpace f)%VS. | Definition | 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... | [
"ahom_in",
"fixedSpace"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomP_tmp {K V f} :
reflect [/\ {in K, forall x, f x = x} &
{in V &, forall x y, f (x * y) = f x * f y}]
(kHom K V f). | Proof.
apply: (iffP andP) => [[/ahom_inP[fM _] /subvP idKf] | [idKf fM]].
by split=> // x /idKf/fixedSpaceP.
split; last by apply/subvP=> x /idKf/fixedSpaceP.
by apply/ahom_inP; split=> //; rewrite idKf ?mem1v.
Qed. | Lemma | kHomP_tmp | 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... | [
"ahom_inP",
"apply",
"fM",
"fixedSpaceP",
"kHom",
"last",
"mem1v",
"split",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomP {K V f} :
reflect [/\ {in V &, forall x y, f (x * y) = f x * f y} &
{in K, forall x, f x = x}]
(kHom K V f). | Proof. by apply: (iffP kHomP_tmp) => [][]. Qed. | Lemma | kHomP | 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",
"kHom",
"kHomP_tmp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAHomP {U V} {f : 'AEnd(L)} :
reflect {in U, forall x, f x = x} (kHom U V f). | Proof. by rewrite /kHom ahomWin; apply: fixedSpacesP. Qed. | Lemma | kAHomP | 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... | [
"ahomWin",
"apply",
"fixedSpacesP",
"kHom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom1 U V : kHom U V \1. | Proof. by apply/kAHomP => u _; rewrite lfunE. Qed. | Lemma | kHom1 | 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",
"kAHomP",
"kHom",
"lfunE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
k1HomE V f : kHom 1 V f = ahom_in V f. | Proof. by apply: andb_idr => /ahom_inP[_ f1]; apply/fixedSpaceP. Qed. | Lemma | k1HomE | 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... | [
"ahom_in",
"ahom_inP",
"apply",
"f1",
"fixedSpaceP",
"kHom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_monoid_morphism (f : 'End(L)) :
reflect (monoid_morphism f) (kHom 1 {:L} f). | Proof. by rewrite k1HomE; apply: ahomP_tmp. Qed. | Lemma | kHom_monoid_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... | [
"ahomP_tmp",
"apply",
"k1HomE",
"kHom",
"monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_lrmorphism (f : 'End(L)) : reflect (multiplicative f) (kHom 1 {:L} f). | Proof. #[warning="-deprecated-since-mathcomp-2.5.0"] by rewrite k1HomE; apply: ahomP. Qed. | Lemma | kHom_lrmorphism | 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... | [
"ahomP",
"apply",
"k1HomE",
"kHom",
"multiplicative"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
k1AHom V (f : 'AEnd(L)) : kHom 1 V f. | Proof. by rewrite k1HomE ahomWin. Qed. | Lemma | k1AHom | 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... | [
"ahomWin",
"k1HomE",
"kHom"
] | Proof. by rewrite k1HomE; apply: ahomP. Qed. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
kHom_poly_id K E f p :
kHom K E f -> p \is a polyOver K -> map_poly f p = p. | Proof.
by case/kHomP_tmp=> idKf _ /polyOverP Kp; apply/polyP=> i; rewrite coef_map /= idKf.
Qed. | Lemma | kHom_poly_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... | [
"apply",
"coef_map",
"kHom",
"kHomP_tmp",
"map_poly",
"polyOver",
"polyOverP",
"polyP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomSl U1 U2 V f : (U1 <= U2)%VS -> kHom U2 V f -> kHom U1 V f. | Proof. by rewrite /kHom => sU12 /andP[-> /(subv_trans sU12)]. Qed. | Lemma | kHomSl | 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... | [
"kHom",
"subv_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomSr K V1 V2 f : (V1 <= V2)%VS -> kHom K V2 f -> kHom K V1 f. | Proof. by move/subvP=> sV12 /kHomP_tmp[idKf /(sub_in2 sV12)fM]; apply/kHomP_tmp. Qed. | Lemma | kHomSr | 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",
"fM",
"kHom",
"kHomP_tmp",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomS K1 K2 V1 V2 f :
(K1 <= K2)%VS -> (V1 <= V2)%VS -> kHom K2 V2 f -> kHom K1 V1 f. | Proof. by move=> sK12 sV12 /(kHomSl sK12)/(kHomSr sV12). Qed. | Lemma | kHomS | 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... | [
"kHom",
"kHomSl",
"kHomSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_eq K E f g :
(K <= E)%VS -> {in E, f =1 g} -> kHom K E f = kHom K E g. | Proof.
move/subvP=> sKE eq_fg; wlog suffices: f g eq_fg / kHom K E f -> kHom K E g.
by move=> IH; apply/idP/idP; apply: IH => x /eq_fg.
case/kHomP_tmp=> idKf fM; apply/kHomP_tmp.
by split=> [x Kx | x y Ex Ey]; rewrite -!eq_fg ?fM ?rpredM // ?idKf ?sKE.
Qed. | Lemma | kHom_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... | [
"apply",
"fM",
"kHom",
"kHomP_tmp",
"rpredM",
"sKE",
"split",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_inv K E f : kHom K E f -> {in E, {morph f : x / x^-1}}. | Proof.
case/kHomP_tmp=> idKf fM x Ex.
have [-> | nz_x] := eqVneq x 0; first by rewrite linear0 invr0 linear0.
have fxV: f x * f x^-1 = 1 by rewrite -fM ?rpredV ?divff // idKf ?mem1v.
have Ufx: f x \is a GRing.unit by apply/unitrPr; exists (f x^-1).
by apply: (mulrI Ufx); rewrite divrr.
Qed. | Lemma | kHom_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... | [
"apply",
"divff",
"divrr",
"eqVneq",
"fM",
"invr0",
"kHom",
"kHomP_tmp",
"linear0",
"mem1v",
"mulrI",
"rpredV",
"unit",
"unitrPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_dim K E f : kHom K E f -> \dim (f @: E) = \dim E. | Proof.
move=> homKf; have [idKf fM] := kHomP_tmp homKf.
apply/limg_dim_eq/eqP; rewrite -subv0; apply/subvP=> v.
rewrite memv_cap memv0 memv_ker => /andP[Ev]; apply: contraLR => nz_v.
by rewrite -unitfE unitrE -(kHom_inv homKf) // -fM ?rpredV ?divff ?idKf ?mem1v.
Qed. | Lemma | kHom_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... | [
"apply",
"dim",
"divff",
"fM",
"kHom",
"kHomP_tmp",
"kHom_inv",
"limg_dim_eq",
"mem1v",
"memv0",
"memv_cap",
"memv_ker",
"rpredV",
"subv0",
"subvP",
"unitfE",
"unitrE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomf : subvs_of E -> L | := f \o vsval. | Let | kHomf | 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... | [
"subvs_of",
"vsval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_is_zmod_morphism : kHom K E f -> zmod_morphism kHomf. | Proof. by case/kHomP_tmp => idKf fM; apply: raddfB. Qed. | Lemma | kHom_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",
"fM",
"kHom",
"kHomP_tmp",
"kHomf",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_is_additive | := kHom_is_zmod_morphism. | Definition | kHom_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... | [
"kHom_is_zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_is_monoid_morphism : kHom K E f -> monoid_morphism kHomf. | Proof.
case/kHomP_tmp=> idKf fM; rewrite /kHomf.
by split=> [|a b] /=; [rewrite algid1 idKf // mem1v | rewrite /= fM ?subvsP].
Qed. | Lemma | kHom_is_monoid_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... | [
"algid1",
"fM",
"kHom",
"kHomP_tmp",
"kHomf",
"mem1v",
"monoid_morphism",
"split",
"subvsP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_is_multiplicative | :=
(fun p => (p.1, p.2)) \o kHom_is_monoid_morphism. | Definition | kHom_is_multiplicative | 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... | [
"kHom_is_monoid_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_rmorphism | := Eval hnf in (kHomf : {rmorphism _ -> _}). | Definition | kHom_rmorphism | 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... | [
"kHomf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_horner K E f p x :
kHom K E f -> p \is a polyOver E -> x \in E -> f p.[x] = (map_poly f p).[f x]. | Proof.
move=> homKf /polyOver_subvs[{}p -> Ex]; pose fRM := kHom_rmorphism homKf.
by rewrite (horner_map _ _ (Subvs Ex)) -[f _](horner_map fRM) map_poly_comp.
Qed. | Lemma | kHom_horner | 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... | [
"horner_map",
"kHom",
"kHom_rmorphism",
"map_poly",
"map_poly_comp",
"polyOver",
"polyOver_subvs"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_root K E f p x :
kHom K E f -> p \is a polyOver E -> x \in E -> root p x ->
root (map_poly f p) (f x). | Proof.
by move/kHom_horner=> homKf Ep Ex /rootP px0; rewrite /root -homKf ?px0 ?raddf0.
Qed. | Lemma | kHom_root | 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... | [
"kHom",
"kHom_horner",
"map_poly",
"polyOver",
"raddf0",
"root",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_root_id K E f p x :
(K <= E)%VS -> kHom K E f -> p \is a polyOver K -> x \in E -> root p x ->
root p (f x). | Proof.
move=> sKE homKf Kp Ex /(kHom_root homKf (polyOverSv sKE Kp) Ex).
by rewrite (kHom_poly_id homKf).
Qed. | Lemma | kHom_root_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... | [
"kHom",
"kHom_poly_id",
"kHom_root",
"polyOver",
"polyOverSv",
"root",
"sKE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomf z | := (map_poly f (Fadjoin_poly E x z)).[y]. | Let | kHomf | 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_poly",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtend_zmod_morphism_subproof : zmod_morphism kHomf. | Proof. by move=> a b; rewrite /kHomf 2!raddfB hornerD hornerN. Qed. | Fact | kHomExtend_zmod_morphism_subproof | 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... | [
"hornerD",
"hornerN",
"kHomf",
"raddfB",
"zmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtend_scalable_subproof : scalable kHomf. | Proof.
move=> k a; rewrite /kHomf linearZ /= -[RHS]mulr_algl -hornerZ; congr _.[_].
by apply/polyP => i; rewrite !(coefZ, coef_map) /= !mulr_algl linearZ.
Qed. | Fact | kHomExtend_scalable_subproof | 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",
"coefZ",
"coef_map",
"hornerZ",
"kHomf",
"linearZ",
"mulr_algl",
"polyP",
"scalable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtendLinear | := Eval hnf in (kHomf : {linear _ -> _}). | Let | kHomExtendLinear | 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... | [
"kHomf",
"linear"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtend | := linfun kHomExtendLinear. | Definition | kHomExtend | 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... | [
"kHomExtendLinear",
"linfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtendE z : kHomExtend z = (map_poly f (Fadjoin_poly E x z)).[y]. | Proof. by rewrite lfunE. Qed. | Lemma | kHomExtendE | 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_poly",
"kHomExtend",
"lfunE",
"map_poly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(sKE : (K <= E)%VS) (homKf : kHom K E f). | Hypotheses | sKE | 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... | [
"kHom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
Px | := (minPoly E x). | Notation | Px | 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... | [
"minPoly"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fPx_y_0 : root (map_poly f Px) y. | Hypothesis | fPx_y_0 | 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",
"map_poly",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
kHomExtend_id z : z \in E -> kHomExtend z = f z. | Proof. by move=> Ez; rewrite kHomExtendE Fadjoin_polyC ?map_polyC ?hornerC. Qed. | Lemma | kHomExtend_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... | [
"Fadjoin_polyC",
"hornerC",
"kHomExtend",
"kHomExtendE",
"map_polyC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtend_val : kHomExtend x = y. | Proof.
have fX: map_poly f 'X = 'X by rewrite (kHom_poly_id homKf) ?polyOverX.
have [Ex | E'x] := boolP (x \in E); last first.
by rewrite kHomExtendE Fadjoin_polyX // fX hornerX.
have:= fPx_y_0; rewrite (minPoly_XsubC Ex) raddfB /= map_polyC fX root_XsubC /=.
by rewrite (kHomExtend_id Ex) => /eqP->.
Qed. | Lemma | kHomExtend_val | 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_polyX",
"fPx_y_0",
"hornerX",
"kHomExtend",
"kHomExtendE",
"kHomExtend_id",
"kHom_poly_id",
"last",
"map_poly",
"map_polyC",
"minPoly_XsubC",
"polyOverX",
"raddfB",
"root_XsubC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtend_poly p :
p \in polyOver E -> kHomExtend p.[x] = (map_poly f p).[y]. | Proof.
move=> Ep; rewrite kHomExtendE (Fadjoin_poly_mod x) //.
rewrite (divp_eq (map_poly f p) (map_poly f Px)).
rewrite !hornerE (rootP fPx_y_0) mulr0 add0r.
have [p1 ->] := polyOver_subvs Ep.
have [Px1 ->] := polyOver_subvs (minPolyOver E x).
by rewrite -map_modp -!map_poly_comp (map_modp (kHom_rmorphism homKf)).
Qed... | Lemma | kHomExtend_poly | 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_poly_mod",
"Px",
"add0r",
"divp_eq",
"fPx_y_0",
"hornerE",
"kHomExtend",
"kHomExtendE",
"kHom_rmorphism",
"map_modp",
"map_poly",
"map_poly_comp",
"minPolyOver",
"mulr0",
"polyOver",
"polyOver_subvs",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHomExtendP : kHom K <<E; x>> kHomExtend. | Proof.
have [idKf fM] := kHomP_tmp homKf.
apply/kHomP_tmp; split=> [z Kz|]; first by rewrite kHomExtend_id ?(subvP sKE) ?idKf.
move=> _ _ /Fadjoin_polyP[p Ep ->] /Fadjoin_polyP[q Eq ->].
rewrite -hornerM !kHomExtend_poly ?rpredM // -hornerM; congr _.[_].
apply/polyP=> i; rewrite coef_map !coefM /= linear_sum /=.
by app... | Lemma | kHomExtendP | 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",
"coefM",
"coef_map",
"eq_bigr",
"fM",
"hornerM",
"kHom",
"kHomExtend",
"kHomExtend_id",
"kHomExtend_poly",
"kHomP_tmp",
"linear_sum",
"polyOverP",
"polyP",
"rpredM",
"sKE",
"split",
"subvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAut U V f | := kHom U V f && (f @: V == V)%VS. | Definition | 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... | [
"kHom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAutE K E f : kAut K E f = kHom K E f && (f @: E <= E)%VS. | Proof.
apply/andP/andP=> [[-> /eqP->] // | [homKf EfE]].
by rewrite eqEdim EfE /= (kHom_dim homKf).
Qed. | Lemma | kAutE | 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",
"eqEdim",
"kAut",
"kHom",
"kHom_dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAutS U1 U2 V f : (U1 <= U2)%VS -> kAut U2 V f -> kAut U1 V f. | Proof. by move=> sU12 /andP[/(kHomSl sU12)homU1f EfE]; apply/andP. Qed. | Lemma | kAutS | 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",
"kAut",
"kHomSl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_kAut_sub K E f : kAut K E f -> kHom K E f. | Proof. by case/andP. Qed. | Lemma | kHom_kAut_sub | 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... | [
"kAut",
"kHom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAut_eq K E (f g : 'End(L)) :
(K <= E)%VS -> {in E, f =1 g} -> kAut K E f = kAut K E g. | Proof.
by move=> sKE eq_fg; rewrite !kAutE (kHom_eq sKE eq_fg) (eq_in_limg eq_fg).
Qed. | Lemma | kAut_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... | [
"eq_in_limg",
"kAut",
"kAutE",
"kHom_eq",
"sKE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAutfE K f : kAut K {:L} f = kHom K {:L} f. | Proof. by rewrite kAutE subvf andbT. Qed. | Lemma | kAutfE | 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... | [
"kAut",
"kAutE",
"kHom",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAut1E E (f : 'AEnd(L)) : kAut 1 E f = (f @: E <= E)%VS. | Proof. by rewrite kAutE k1AHom. Qed. | Lemma | kAut1E | 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... | [
"k1AHom",
"kAut",
"kAutE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAutf_lker0 K f : kHom K {:L} f -> lker f == 0%VS. | Proof.
move/(kHomSl (sub1v _))/kHom_monoid_morphism => fM.
pose fmM := GRing.isMonoidMorphism.Build _ _ _ fM.
pose fRM : {rmorphism _ -> _} := HB.pack (fun_of_lfun f) fmM.
by apply/lker0P; apply: (fmorph_inj fRM).
Qed. | Lemma | kAutf_lker0 | 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... | [
"Build",
"apply",
"fM",
"fmorph_inj",
"fun_of_lfun",
"kHom",
"kHomSl",
"kHom_monoid_morphism",
"lker",
"lker0P",
"sub1v"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_kHomf K f : kHom K {:L} f -> kHom K {:L} f^-1. | Proof.
move=> homKf; have [[idKf fM] kerf0] := (kHomP_tmp homKf, kAutf_lker0 homKf).
have f1K: cancel f^-1%VF f by apply: lker0_lfunVK.
apply/kHomP_tmp; split=> [x Kx | x y _ _]; apply: (lker0P kerf0).
by rewrite f1K idKf.
by rewrite fM ?memvf ?{1}f1K.
Qed. | Lemma | inv_kHomf | 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",
"fM",
"kAutf_lker0",
"kHom",
"kHomP_tmp",
"kerf0",
"lker0P",
"lker0_lfunVK",
"memvf",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_is_ahom (f : 'AEnd(L)) : ahom_in {:L} f^-1. | Proof.
have /ahomP_tmp/kHom_monoid_morphism hom1f := valP f.
exact/ahomP_tmp/kHom_monoid_morphism/inv_kHomf.
Qed. | Lemma | inv_is_ahom | 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... | [
"ahomP_tmp",
"ahom_in",
"inv_kHomf",
"kHom_monoid_morphism",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_ahom (f : 'AEnd(L)) : 'AEnd(L) | := AHom (inv_is_ahom f). | Canonical | inv_ahom | 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... | [
"inv_is_ahom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"f ^-1" | := (inv_ahom f) : lrfun_scope. | Notation | f ^-1 | 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... | [
"inv_ahom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_kHom_img K E f g :
kHom K (g @: E) f -> kHom K E g -> kHom K E (f \o g). | Proof.
move=> /kHomP_tmp[idKf fM] /kHomP_tmp[idKg gM]; apply/kHomP_tmp; split=> [x Kx | x y Ex Ey].
by rewrite lfunE /= idKg ?idKf.
by rewrite !lfunE /= gM // fM ?memv_img.
Qed. | Lemma | comp_kHom_img | 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",
"fM",
"kHom",
"kHomP_tmp",
"lfunE",
"memv_img",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_kHom K E f g : kHom K {:L} f -> kHom K E g -> kHom K E (f \o g). | Proof. by move/(kHomSr (subvf (g @: E))); apply: comp_kHom_img. Qed. | Lemma | comp_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... | [
"apply",
"comp_kHom_img",
"kHom",
"kHomSr",
"subvf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_extends K E f p U :
(K <= E)%VS -> kHom K E f ->
p \is a polyOver K -> splittingFieldFor E p U ->
{g | kHom K U g & {in E, f =1 g}}. | Proof.
move=> sKE homEf Kp /sig2_eqW[rs Dp <-{U}].
set r := rs; have rs_r: all [in rs] r by apply/allP.
elim: r rs_r => [_|z r IHr /=/andP[rs_z rs_r]] /= in E f sKE homEf *.
by exists f; rewrite ?Fadjoin_nil.
set Ez := <<E; z>>%AS; pose fpEz := map_poly f (minPoly E z).
suffices{IHr} /sigW[y fpEz_y]: exists y, root f... | Lemma | kHom_extends | 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_nil",
"adjoin_cons",
"all",
"allP",
"apply",
"dvdp_map",
"dvdp_prod_XsubC",
"eqp_dvdr",
"eqp_root",
"eqp_size",
"id",
"kHom",
"kHomExtend",
"kHomExtendP",
"kHomExtend_id",
"kHomP_tmp",
"kHom_poly_id",
"kHom_rmorphism",
"lead_coefE",
"ltnS",
"map_poly",
"map_poly_co... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitting_field_axiom (F : fieldType) (L : fieldExtType F) | :=
exists2 p : {poly L}, p \is a polyOver 1%VS & splittingFieldFor 1 p {:L}. | Definition | splitting_field_axiom | 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... | [
"poly",
"polyOver",
"splittingFieldFor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_field_splitting (F : fieldType) (L : fieldExtType F) :
(forall (K : {subfield L}) x,
exists r, minPoly K x == \prod_(y <- r) ('X - y%:P)) ->
SplittingField.axiom L. | Proof.
move=> normalL; pose r i := sval (sigW (normalL 1%AS (tnth (vbasis {:L}) i))).
have sz_r i: size (r i) <= \dim {:L}.
rewrite -ltnS -(size_prod_XsubC _ id) /r; case: sigW => _ /= /eqP <-.
rewrite size_minPoly ltnS; move: (tnth _ _) => x.
by rewrite adjoin_degreeE dimv1 divn1 dimvS // subvf.
pose mkf (z : L)... | Lemma | normal_field_splitting | 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_degreeE",
"apply",
"axiom",
"big_filter",
"big_map",
"big_mkord",
"big_nth",
"big_ord_narrow",
"dim",
"dimv1",
"dimvS",
"divn1",
"eqEsubv",
"eqpxx",
"id",
"in_tuple",
"ltnS",
"ltn_ord",
"map",
"mapP",
"mem_filter",
"mem_index_enum",
"minPoly",
"minPolyOver",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
regular_splittingAxiom (F : fieldType) : SplittingField.axiom F^o. | Proof.
exists 1; first exact: rpred1.
by exists [::]; [rewrite big_nil eqpxx | rewrite Fadjoin_nil regular_fullv].
Qed. | Fact | regular_splittingAxiom | 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_nil",
"axiom",
"big_nil",
"eqpxx",
"regular_fullv",
"rpred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splittingFieldP : SplittingField.axiom L. | Proof. exact: splittingFieldP_subproof. Qed. | Lemma | splittingFieldP | 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... | [
"axiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splittingPoly :
{p : {poly L} | p \is a polyOver 1%VS & splittingFieldFor 1 p {:L}}. | Proof.
pose factF p s := (p \is a polyOver 1%VS) && (p %= \prod_(z <- s) ('X - z%:P)).
suffices [[p rs] /andP[]]: {ps | factF F L ps.1 ps.2 & <<1 & ps.2>> = {:L}}%VS.
by exists p; last exists rs.
apply: sig2_eqW; have [p F0p [rs splitLp genLrs]] := splittingFieldP.
by exists (p, rs); rewrite // /factF F0p splitLp.
Qe... | Lemma | splittingPoly | 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",
"last",
"poly",
"polyOver",
"sig2_eqW",
"splittingFieldFor",
"splittingFieldP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fieldOver_splitting E : SplittingField.axiom (fieldOver E). | Proof.
have [p Fp [r Dp defL]] := splittingFieldP; exists p.
apply/polyOverP=> j; rewrite trivial_fieldOver.
by rewrite (subvP (sub1v E)) ?(polyOverP Fp).
exists r => //; apply/vspaceP=> x; rewrite memvf.
have [L0 [_ _ defL0]] := @aspaceOverP _ _ E <<1 & r : seq (fieldOver E)>>.
rewrite defL0; have: x \in <<1 & r>... | Fact | fieldOver_splitting | 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_seqP",
"L0",
"apply",
"aspaceOverP",
"axiom",
"fieldOver",
"mem1v",
"memvE",
"memvf",
"polyOverP",
"seq",
"seqv_sub_adjoin",
"split",
"splittingFieldP",
"sub1v",
"subvP",
"trivial_fieldOver",
"vspaceP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
enum_AEnd : {kAutL : seq 'AEnd(L) | forall f, f \in kAutL}. | Proof.
pose isAutL (s : seq 'AEnd(L)) (f : 'AEnd(L)) := kHom 1 {:L} f = (f \in s).
suffices [kAutL in_kAutL] : {kAutL : seq 'AEnd(L) | forall f, isAutL kAutL f}.
by exists kAutL => f; rewrite -in_kAutL k1AHom.
have [p Kp /sig2_eqW[rs Dp defL]] := splittingPoly.
do [rewrite {}/isAutL -(erefl (asval 1)); set r := rs; s... | Lemma | enum_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... | [
"Build",
"Fadjoin_nil",
"Fadjoin_polyP",
"adjoin_cons",
"adjoin_degreeE",
"adjoin_seqSr",
"all",
"allP",
"allpairsP",
"apply",
"comp_kHom",
"dvdp_prod_XsubC",
"enum",
"eqEsubv",
"eqSS",
"eqp_dvdr",
"eqp_root",
"eqp_size",
"f1",
"fun_of_lfun",
"horner_map",
"id_lfunE",
"in... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
splitting_field_normal K x :
exists r, minPoly K x == \prod_(y <- r) ('X - y%:P). | Proof.
pose q1 := minPoly 1 x; pose fx_root q (f : 'AEnd(L)) := root q (f x).
have [[p F0p splitLp] [autL DautL]] := (splittingFieldP, enum_AEnd).
suffices{K} autL_px q: q != 0 -> q %| q1 -> size q > 1 -> has (fx_root q) autL.
set q := minPoly K x; have: q \is monic := monic_minPoly K x.
have: q %| q1 by rewrite mi... | Lemma | splitting_field_normal | 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... | [
"Build",
"FadjoinP",
"Fadjoin_nil",
"Fadjoin_polyP",
"Fadjoin_seqP",
"addn2",
"adjoin_rcons",
"adjoin_seqSl",
"adjoin_seqSr",
"ahom_in",
"ahom_inP",
"aimg1",
"all",
"allP",
"all_rcons",
"apply",
"aspace",
"baseFieldType",
"big_cons",
"big_map",
"big_nil",
"coefK",
"coef_m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kHom_to_AEnd K E f : kHom K E f -> {g : 'AEnd(L) | {in E, f =1 val g}}. | Proof.
move=> homKf; have{homKf} [homFf sFE] := (kHomSl (sub1v K) homKf, sub1v E).
have [p Fp /(splittingFieldForS sFE (subvf E))splitLp] := splittingPoly.
have [g0 homLg0 eq_fg] := kHom_extends sFE homFf Fp splitLp.
by apply: exist (Sub g0 _) _ => //; apply/ahomP_tmp/kHom_monoid_morphism.
Qed. | Lemma | kHom_to_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... | [
"Sub",
"ahomP_tmp",
"apply",
"kHom",
"kHomSl",
"kHom_extends",
"kHom_monoid_morphism",
"splittingFieldForS",
"splittingPoly",
"sub1v",
"subvf",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inAEnd f | := SeqSub (svalP (enum_AEnd L) f). | Definition | inAEnd | 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... | [
"enum_AEnd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inAEndK : cancel inAEnd val. | Proof. by []. Qed. | Fact | inAEndK | 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... | [
"inAEnd",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_AEnd (f g : 'AEnd(L)) : 'AEnd(L) | := (g \o f)%AF. | Definition | comp_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... | [] | the group operation is the categorical composition operation | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
comp_AEndA : associative comp_AEnd. | Proof. by move=> f g h; apply: val_inj; symmetry; apply: comp_lfunA. Qed. | Fact | comp_AEndA | 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_AEnd",
"comp_lfunA",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_AEnd1l : left_id \1%AF comp_AEnd. | Proof. by move=> f; apply/val_inj/comp_lfun1r. Qed. | Fact | comp_AEnd1l | 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_AEnd",
"comp_lfun1r",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comp_AEndK : left_inverse \1%AF (@inv_ahom _ L) comp_AEnd. | Proof. by move=> f; apply/val_inj; rewrite /= lker0_compfV ?AEnd_lker0. Qed. | Fact | comp_AEndK | 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",
"comp_AEnd",
"inv_ahom",
"lker0_compfV",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAEnd U V | := [set f : 'AEnd(L) | kAut U V f]. | Definition | kAEnd | 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... | [
"kAut"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAEndf U | := kAEnd U {:L}. | Definition | kAEndf | 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... | [
"kAEnd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAEnd_group_set K E : group_set (kAEnd K E). | Proof.
apply/group_setP; split=> [|f g]; first by rewrite inE /kAut kHom1 lim1g eqxx.
rewrite !inE !kAutE => /andP[homKf EfE] /andP[/(kHomSr EfE)homKg EgE].
by rewrite (comp_kHom_img homKg homKf) limg_comp (subv_trans _ EgE) ?limgS.
Qed. | Lemma | kAEnd_group_set | 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_kHom_img",
"eqxx",
"group_set",
"group_setP",
"inE",
"kAEnd",
"kAut",
"kAutE",
"kHom1",
"kHomSr",
"lim1g",
"limgS",
"limg_comp",
"split",
"subv_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAEnd_group K E | := group (kAEnd_group_set K E). | Canonical | kAEnd_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... | [
"group",
"kAEnd_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kAEndf_group K | := [group of kAEndf K]. | Canonical | kAEndf_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... | [
"group",
"kAEndf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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