(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
(******************************************************************************)
(*   The main result in this file is the existence theorem that underpins the *)
(* construction of the algebraic numbers in file algC.v. This theorem simply  *)
(* asserts the existence of an algebraically closed field with an             *)
(* automorphism of order 2, and dubbed the Fundamental_Theorem_of_Algebraics  *)
(* because it is essentially the Fundamental Theorem of Algebra for algebraic *)
(* numbers (the more familiar version for complex numbers can be derived by   *)
(* continuity).                                                               *)
(*   Although our proof does indeed construct exactly the algebraics, we      *)
(* choose not to expose this in the statement of our Theorem. In algC.v we    *)
(* construct the norm and partial order of the "complex field" introduced by  *)
(* the Theorem; as these imply is has characteristic 0, we then get the       *)
(* algebraics as a subfield. To avoid some duplication a few basic properties *)
(* of the algebraics, such as the existence of minimal polynomials, that are  *)
(* required by the proof of the Theorem, are also proved here.                *)
(*  The main theorem of closed_field supplies us directly with an algebraic   *)
(* closure of the rationals (as the rationals are a countable field), so all  *)
(* we really need to construct is a conjugation automorphism that exchanges   *)
(* the two roots (i and -i) of X^2 + 1, and fixes a (real) subfield of        *)
(* index 2. This does not require actually constructing this field: the       *)
(* kHomExtend construction from galois.v supplies us with an automorphism     *)
(* conj_n of the number field Q[z_n] = Q[x_n, i] for any x_n such that Q[x_n] *)
(* does not contain i (e.g., such that Q[x_n] is real). As conj_n will extend *)
(* conj_m when Q[x_n] contains x_m, it therefore suffices to construct a      *)
(* sequence x_n such that                                                     *)
(* (1) For each n, Q[x_n] is a REAL field containing Q[x_m] for all m <= n.   *)
(* (2) Each z in C belongs to Q[z_n] = Q[x_n, i] for large enough n.          *)
(* This, of course, amounts to proving the Fundamental Theorem of Algebra.    *)
(*   Indeed, we use a constructive variant of Artin's algebraic proof of that *)
(* Theorem to replace (2) by                                                  *)
(* (3) Each monic polynomial over Q[x_m] whose constant term is -c^2 for some *)
(*     c in Q[x_m] has a root in Q[x_n] for large enough n.                   *)
(* We then ensure (3) by setting Q[x_n+1] = Q[x_n, y] where y is the root of  *)
(* of such a polynomial p found by dichotomy in some interval [0, b] with b   *)
(* suitably large (such that p[b] >= 0), and p is obtained by decoding n into *)
(* a triple (m, p, c) that satisfies the conditions of (3) (taking x_n+1=x_n  *)
(* if this is not the case), thereby ensuring that all such triples are       *)
(* ultimately considered.                                                     *)
(*   In more detail, the 600-line proof consists in six (uneven) parts:       *)
(* (A) - Construction of number fields (~ 100 lines): in order to make use of *)
(*     the theory developped in falgebra, fieldext, separable and galois we   *)
(*     construct a separate fielExtType Q z for the number field Q[z], with   *)
(*     z in C, the closure of rat supplied by countable_algebraic_closure.    *)
(*     The morphism (ofQ z) maps Q z to C, and the Primitive Element Theorem  *)
(*     lets us define a predicate sQ z characterizing the image of (ofQ z),   *)
(*     as well as a partial inverse (inQ z) to (ofQ z).                       *)
(* (B) - Construction of the real extension Q[x, y] (~ 230 lines): here y has *)
(*     to be a root of a polynomial p over Q[x] satisfying the conditions of  *)
(*     (3), and Q[x] should be real and archimedean, which we represent by    *)
(*     a morphism from Q x to some archimedean field R, as the ssrnum and     *)
(*     fieldext structures are not compatible. The construction starts by     *)
(*     weakening the condition p[0] = -c^2 to p[0] <= 0 (in R), then reducing *)
(*     to the case where p is the minimal polynomial over Q[x] of some y (in  *)
(*     some Q[w] that contains x and all roots of p). Then we only need to    *)
(*     construct a realFieldType structure for Q[t] = Q[x,y] (we don't even   *)
(*     need to show it is consistent with that of R). This amounts to fixing  *)
(*     the sign of all z != 0 in Q[t], consistently with arithmetic in Q[t].  *)
(*     Now any such z is equal to q[y] for some q in Q[x][X] coprime with p.  *)
(*     Then up + vq = 1 for Bezout coefficients u and v. As p is monic, there *)
(*     is some b0 >= 0 in R such that p changes sign in ab0 = [0; b0]. As R   *)
(*     is archimedean, some iteration of the binary search for a root of p in *)
(*     ab0 will yield an interval ab_n such that |up[d]| < 1/2 for d in ab_n. *)
(*     Then |q[d]| > 1/2M > 0 for any upper bound M on |v[X]| in ab0, so q    *)
(*     cannot change sign in ab_n (as then root-finding in ab_n would yield a *)
(*     d with |Mq[d]| < 1/2), so we can fix the sign of z to that of q in     *)
(*     ab_n.                                                                  *)
(* (C) - Construction of the x_n and z_n (~50 lines): x_ n is obtained by     *)
(*     iterating (B), starting with x_0 = 0, and then (A) and the PET yield   *)
(*     z_ n. We establish (1) and (3), and that the minimal polynomial of the *)
(*     preimage i_ n of i over the preimage R_ n of Q[x_n] is X^2 + 1.        *)
(* (D) - Establish (2), i.e., prove the FTA (~180 lines). We must depart from *)
(*     Artin's proof because deciding membership in the union of the Q[x_n]   *)
(*     requires the FTA, i.e., we cannot (yet) construct a maximal real       *)
(*     subfield of C. We work around this issue by first reducing to the case *)
(*     where Q[z] is Galois over Q and contains i, then using induction over  *)
(*     the degree of z over Q[z_ n] (i.e., the degree of a monic polynomial   *)
(*     over Q[z_n] that has z as a root). We can assume that z is not in      *)
(*     Q[z_n]; then it suffices to find some y in Q[z_n, z] \ Q[z_n] that is  *)
(*     also in Q[z_m] for some m > n, as then we can apply induction with the *)
(*     minimal polynomial of z over Q[z_n, y]. In any Galois extension Q[t]   *)
(*     of Q that contains both z and z_n, Q[x_n, z] = Q[z_n, z] is Galois     *)
(*     over both Q[x_n] and Q[z_n]. If Gal(Q[x_n,z] / Q[x_n]) isn't a 2-group *)
(*     take one of its Sylow 2-groups P; the minimal polynomial p of any      *)
(*     generator of the fixed field F of P over Q[x_n] has odd degree, hence  *)
(*     by (3) - p[X]p[-X] and thus p has a root y in some Q[x_m], hence in    *)
(*     Q[z_m]. As F is normal, y is in F, with minimal polynomial p, and y    *)
(*     is not in Q[z_n] = Q[x_n, i] since p has odd degree. Otherwise,        *)
(*     Gal(Q[z_n,z] / Q[z_n]) is a proper 2-group, and has a maximal subgroup *)
(*     P of index 2. The fixed field F of P has a generator w over Q[z_n]     *)
(*     with w^2 in Q[z_n] \ Q[x_n], i.e. w^2 = u + 2iv with v != 0. From (3)  *)
(*     X^4 - uX^2 - v^2 has a root x in some Q[x_m]; then x != 0 as v != 0,   *)
(*     hence w^2 = y^2 for y = x + iv/x in Q[z_m], and y generates F.         *)
(* (E) - Construct conj and conclude (~40 lines): conj z is defined as        *)
(*     conj_ n z with the n provided by (2); since each conj_ m is a morphism *)
(*     of order 2 and conj z = conj_ m z for any m >= n, it follows that conj *)
(*     is also a morphism of order 2.                                         *)
(* Note that (C), (D) and (E) only depend on Q[x_n] not containing i; the     *)
(* order structure is not used (hence we need not prove that the ordering of  *)
(* Q[x_m] is consistent with that of Q[x_n] for m >= n).                      *)
(******************************************************************************)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Notation "p ^ f" := (map_poly f p) : ring_scope.
Local Notation "p ^@" := (p ^ in_alg _) (at level 2, format "p ^@"): ring_scope.
Local Notation "<< E ; u >>" := <<E; u>>%VS.
Local Notation Qmorphism C := {rmorphism rat -> C}.
Definition decidable_embedding sT T (f : sT -> T) :=
forall y, decidable (exists x, y = f x).
Prenex Implicits alg_integral.
Arguments map_poly_inj {F R} f [p1 p2].

Theorem Fundamental_Theorem_of_Algebraics :
{L : closedFieldType &
{conj : {rmorphism L -> L} | involutive conj & ~ conj =1 id}}.