On pseudo algebraically closed extensions of fields

Abstract

The notion of 'Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. As a result we get a classification of all finite PAC extensions which in turn proves the 'bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g. transitiveness of PAC extensions.

Projective pairs of profinite groups

“…A C double embedding problem, or in short C-DEP, for the pair (Γ, Λ) is a commutative diagram (1) where G, H, A, B ∈ C, A B, G H , i, j, ϕ are the inclusion maps, and α, μ, β, ν are surjective. Therefore a C-DEP consists of two compatible C-EPs: the lower embedding problem (μ, α) for Γ and the higher embedding problem (ν, β) for Λ.…”

“…In the above theorem we actually prove that for almost all σ ∈ Λ e the pair ( σ , Λ) has the following stronger lifting property. For any C-EP (1) and for any h ∈ G e that satisfies α(h) = μ(σ )…”

“…By [1,Theorem 4.2] we can assume that K /K 0 is a separable algebraic extension. By Corollary 3.3 P embeds into Gal(K ) in such a way that (P , Gal(K )) is projective.…”

“…We have to prove that M/K is PAC.Assume that M/K is also separable. We use[1, Proposition 4.5] which says that it suffices to geometrically solve (in the weak sense) each finite rational double embedding problem. Since…”

Fields on the Bottom

“…We mention that Lior Bary-Soroker [1] strengthened the bottom theorem in the following way: Let K be a finitely generated extension of Q and let e ≥ 2 be an integer. Then, for almost all (σ 1 , .…”