On pseudo algebraically closed extensions of fields

On Pseudo Algebraically Closed Extensions of Fields

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....

Pseudo Algebraically Closed Extensions

Pseudo Algebraically Closed Fields over Rings

We prove that for almost all σ ∈ G(Q)e the field Q̃(σ) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating separable rational map φ: V → Ar there exists a point a ∈ V (Q̃(σ)) such that φ(a) ∈ Zr. We then say that Q̃(σ) is PAC over Z. This is a stronger property then being PAC. Indeed we show that beside the fields Q̃(σ) other fields which ...

PSEUDO ALGEBRAICALLY CLOSED FIELDS OVER RINGSbyMoshe

We prove that for almost all 2 G(Q) e the eld ~ Q() has the following property: For each absolutely irreducible aane variety V of dimension r and each dominating separable rational map ': V ! A r there exists a point a 2 V (~ Q()) such that '(a) 2 Z r. We then say that ~ Q() is PAC over Z. This is a stronger property then being PAC. Indeed we show that beside the elds ~ Q() other elds which are...

Deenability of Geometric Properties in Algebraically Closed Fields Deenability of Geometric Properties in Algebraically Closed Fields Deenability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I satisfying the following property for every de nable set X C X is connected if and only if C X j F where I is interpreted by X We conjecture that the same result holds for the closed subsets of C We prove some results motivated by this conjecture