Pseudo Algebraically Closed Extensions

TL;DR

This PhD explores pseudo algebraically closed (PAC) extensions of fields, introducing a group-theoretic framework and revealing that the Galois closure of a proper PAC extension is separably closed, with applications to number theory.

Contribution

It develops a new group-theoretic approach to study PAC extensions and establishes key properties of their Galois closures, expanding understanding of their structure.

Findings

Galois closure of a proper PAC extension is separably closed

Introduces the concept of projective pairs as a group-theoretic counterpart

Provides an analog of Dirichlet's theorem for polynomial rings over infinite fields

Abstract

This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.