Lior Bary-Soroker

Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x, y)) have free absolute Galois groups.

Dirichlet's theorem for polynomial rings

We prove the following form of Dirichlets theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F. For all relatively prime polynomials a(X), b(X) E F[X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) E F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.

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.

On the function field analogue of Landau's theorem on sums of squares

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in

Fully Hilbertian fields

\mathbb{Z}

Is a bivariate polynomial with ± 1 coefficients irreducible? Very likely!

. We define the analogue of a sum of two squares in

On the Characterization of Hilbertian Fields

\mathbb{F}_q[T]

Subgroup Structure of Fundamental Groups in Positive Characteristic

and estimate the number

On fields of totally

B_q(n)

\mathfrak{S}

of such polynomials of degree