From mathcomp Require Import all_ssreflect all_fingroup all_algebra. From mathcomp Require Import all_solvable all_field. From Abel Require Import char0 various. Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import GRing.Theory. Local Open Scope ring_scope. Lemma Cyclotomic1 : 'Phi_1 = 'X - 1. Proof. by have := @prod_Cyclotomic 1%N isT; rewrite big_cons big_nil mulr1. Qed. Lemma Cyclotomic2 : 'Phi_2 = 'X + 1. Proof. have := @prod_Cyclotomic 2%N isT; rewrite !big_cons big_nil mulr1/=. rewrite Cyclotomic1 -(@expr1n [ringType of {poly int}] 2%N). by rewrite subr_sqr expr1n => /mulfI->//; rewrite polyXsubC_eq0. Qed. Lemma prim_root1 (F : fieldType) n : (n.-primitive_root (1 : F)) = (n == 1)%N. Proof. case: n => [|[|n]]//. by apply/'forall_eqP => i; rewrite ord1//= eqxx; apply/unity_rootP. apply/'forall_eqP => /= /(_ (@Ordinal _ n _))/=/(_ _)/unity_rootP. by rewrite !ltnS leqnSn ltn_eqF//; apply => //; rewrite expr1n. Qed. Lemma prim2_rootN1 (F : fieldType) : 2%:R != 0 :> F -> 2.-primitive_root (- 1 : F). Proof. move=> tow_neq0; apply/'forall_eqP => -[[|[|]]]//= _; last first. by apply/unity_rootP; rewrite -signr_odd. by apply/unity_rootP/eqP; rewrite expr1 eq_sym -addr_eq0 -mulr2n. Qed. Section PhiCyclotomic. Variable (F : fieldType). Local Notation ZtoF := (intr : int -> F). Local Notation pZtoF := (map_poly ZtoF). Lemma Phi_cyclotomic (n : nat) (w : F) : n.-primitive_root w -> pZtoF 'Phi_n = cyclotomic w n. Proof. elim/ltn_ind: n w => n ihn w prim_w. have n_gt0 := prim_order_gt0 prim_w. pose P k := pZtoF 'Phi_k. pose Q k := cyclotomic (w ^+ (n %/ k)) k. have eP : \prod_(d <- divisors n) P d = 'X^n - 1. by rewrite -rmorph_prod /= prod_Cyclotomic // rmorphB /= map_polyC map_polyXn. have eQ : \prod_(d <- divisors n) Q d = 'X^n - 1 by rewrite -prod_cyclotomic. have fact (u : nat -> {poly F}) : \prod_(d <- divisors n) u d = u n * \prod_(d <- rem n (divisors n)) u d. by rewrite [LHS](big_rem n) ?divisors_id. pose p := \prod_(d <- rem n (divisors n)) P d. pose q := \prod_(d <- rem n (divisors n)) Q d. have ePp : P n * p = 'X^n - 1 by rewrite -eP fact. have eQq : Q n * q = 'X^n - 1 by rewrite -eQ fact. have Xnsub1N0 : 'X^n - 1 != 0 :> {poly F}. by rewrite -size_poly_gt0 size_Xn_sub_1. have pN0 : p != 0 by apply: dvdpN0 Xnsub1N0; rewrite -ePp dvdp_mulIr. have epq : p = q. case: (divisors_correct n_gt0) => uniqd sortedd dP. apply: eq_big_seq=> i; rewrite mem_rem_uniq ?divisors_uniq // inE. case/andP=> NiSn di; apply: ihn; last by apply: dvdn_prim_root; rewrite -?dP. suff: (i <= n)%N by rewrite leq_eqVlt (negPf NiSn). by apply: dvdn_leq => //; rewrite -dP. have {epq} : P n * p = Q n * p by rewrite [in RHS]epq ePp eQq. by move/(mulIf pN0); rewrite /Q divnn n_gt0. Qed. End PhiCyclotomic. Section CyclotomicExt. Variables (F0 : fieldType) (L : fieldExtType F0). Variables (E : {subfield L}) (w : L) (n : nat). Hypothesis w_is_nth_root : n.-primitive_root w. Lemma splitting_Fadjoin_cyclotomic : splittingFieldFor E (cyclotomic w n) <>. Proof. exists [seq w ^+ val k | k <- enum 'I_n & coprime (val k) n]. by rewrite /cyclotomic big_map big_filter big_enum_cond/= eqpxx. rewrite map_comp -(filter_map _ (fun i => coprime i n)) val_enum_ord. have [n_gt1|] := ltnP 1 n; last first. case: n w_is_nth_root (prim_order_gt0 w_is_nth_root) => [|[|]]//= wnth _ _. by rewrite adjoin_seq1 expr0 -[w]expr1 prim_expr_order. set s := (X in <<_ & X>>%VS); suff /eq_adjoin-> : s =i w :: s. rewrite adjoin_cons (Fadjoin_seq_idP _)//. by apply/allP => _/mapP[i _ ->]/=; rewrite rpredX// memv_adjoin. move=> x; rewrite in_cons orbC; symmetry; have []//= := boolP (_ \in _). apply: contraNF => /eqP ->; rewrite -[w]expr1 map_f//. by rewrite mem_filter mem_iota// coprime1n. Qed. Lemma cyclotomic_over : cyclotomic w n \is a polyOver E. Proof. by apply/polyOverP=> i; rewrite -Phi_cyclotomic // coef_map /= rpred_int. Qed. Hint Resolve cyclotomic_over : core. End CyclotomicExt. Section Cyclotomic. (* MISSING *) Lemma primitive_root_pow (F : fieldType) (m : nat) (w w' : F) : m.-primitive_root w' -> m.-primitive_root w -> exists2 k, coprime k m & w = w' ^+ k. Proof. move/root_cyclotomic<-. rewrite /cyclotomic -big_filter; have [t et [uniqs tP /= perms]] := big_enumP. pose rs := [seq w' ^+ (val i) | i <- t]; set p := (X in root X). have {p} -> : p = \prod_(w <- rs) ('X - w%:P) by rewrite /p big_map. rewrite root_prod_XsubC; case/mapP=> [[i ltim]]; rewrite tP /= => coprim ew. by exists i. Qed. Variables (F0 : fieldType) (L : splittingFieldType F0). Variables (E : {subfield L}) (w : L) (n : nat). Hypothesis w_is_nth_root : n.-primitive_root w. (** Easy **) (* - E(x) is Galois *) Lemma galois_Fadjoin_cyclotomic : galois E <>. Proof. apply/splitting_galoisField; exists (cyclotomic w n). split; rewrite ?cyclotomic_over//; last exact: splitting_Fadjoin_cyclotomic. rewrite /cyclotomic -(big_image _ _ _ (fun x => 'X - x%:P))/=. rewrite separable_prod_XsubC map_inj_uniq ?enum_uniq// => i j /eqP. by rewrite (eq_prim_root_expr w_is_nth_root) !modn_small// => /eqP/val_inj. Qed. Lemma abelian_cyclotomic : abelian 'Gal(<> / E)%g. Proof. case: (boolP (w \in E)) => [w_in_E |w_notin_E]. suff -> : ('Gal(<> / E) = 1)%g by apply: abelian1. apply/eqP; rewrite -subG1; apply/subsetP => x x_in. rewrite inE gal_adjoin_eq ?group1 // (fixed_gal _ x_in w_in_E) ?gal_id //. by have /Fadjoin_idP H := w_in_E; rewrite -{1}H subvv. rewrite card_classes_abelian /classes. apply/eqP; apply: card_in_imset => f g f_in g_in; rewrite -!orbitJ. move/orbit_eqP/orbitP => [] h h_in <- {f f_in}; apply/eqP. rewrite gal_adjoin_eq //= /conjg /= ?groupM ?groupV //. rewrite ?galM ?memv_gal ?memv_adjoin //. have hg_gal f : f \in 'Gal(<> / E)%g -> f w ^+ n = 1. by move=> f_in; apply/prim_expr_order; rewrite fmorph_primitive_root. have := svalP (prim_rootP w_is_nth_root (hg_gal _ g_in)). have h1_in : (h ^-1)%g \in 'Gal(<> / E)%g by rewrite ?groupV. have := svalP (prim_rootP w_is_nth_root (hg_gal _ h1_in)). set ih1 := sval _ => hh1; set ig := sval _ => hg. rewrite hh1 rmorphX /= hg exprAC -hh1 rmorphX /=. by rewrite -galM ?memv_adjoin // mulVg gal_id. Qed. (* - Gal(E(x) / E) is then solvable *) Lemma solvable_Fadjoin_cyclotomic : solvable 'Gal(<> / E). Proof. exact/abelian_sol/abelian_cyclotomic. Qed. End Cyclotomic.