(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* This file provides some standard results based on integrality properties   *)
(* of characters, such as theorem asserting that the degree of an irreducible *)
(* character of G divides the order of G (Isaacs 3.11), or the famous p^a.q^b *)
(* solvability theorem of Burnside.                                           *)
(*   Defined here:                                                            *)
(*    'K_k == the kth class sum in gring F G, where k : 'I_#|classes G|, and  *)
(*            F is inferred from the context.                                 *)
(*            := gset_mx F G (enum_val k) (see mxrepresentation.v).           *)
(* --> The 'K_k form a basis of 'Z(group_ring F G)%MS.                        *)
(*   gring_classM_coef i j k == the coordinate of 'K_i *m 'K_j on 'K_k; this  *)
(*            is usually abbreviated as a i j k.                              *)
(*   gring_classM_coef_set A B z == the set of all (x, y) in setX A B such    *)
(*            that x * y = z; if A and B are respectively the ith and jth     *)
(*            conjugacy class of G, and z is in the kth conjugacy class, then *)
(*            gring_classM_coef i j k is exactly the cardinal of this set.    *)
(*  'omega_i[A] == the mode of 'chi[G]_i on (A \in 'Z(group_ring algC G))%MS, *)
(*            i.e., the z such that gring_op 'Chi_i A  = z%:M.                *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.

(* Specifications: *)
Lemma group_num_field_exists (gT : finGroupType) (G : {group gT}) :
{Qn : splittingFieldType rat & galois 1 {:Qn} &
{QnC : {rmorphism Qn -> algC}
& forall nuQn : argumentType (mem ('Gal({:Qn}%VS / 1%VS))),
{nu : {rmorphism algC -> algC} |
{morph QnC: a / nuQn a >-> nu a}}
& {w : Qn & #|G|.-primitive_root w /\ <<1; w>>%VS = fullv
& forall (hT : finGroupType) (H : {group hT}) (phi : 'CF(H)),
phi \is a character ->
forall x, (#[x] %| #|G|)%N -> {a | QnC a = phi x}}}}.