PAC Fields over Finitely Generated Fields

TL;DR

This paper proves that for finitely generated fields, any non-separably closed Galois extension cannot be pseudo-algebraically closed over the base field, clarifying the structure of such extensions.

Contribution

It establishes a new theorem characterizing PAC extensions over finitely generated fields, specifically excluding non-separably closed Galois extensions.

Findings

Galois extensions of finitely generated fields are not PAC if not separably closed

Provides a clear criterion for PAC extensions over finitely generated fields

Advances understanding of the structure of Galois and PAC extensions

Abstract

We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$.