On the transcendental Galois extensions

TL;DR

This paper introduces and compares various types of transcendental Galois extensions of fields, highlighting their properties, differences from algebraic Galois extensions, and specific cases like purely transcendental extensions.

Contribution

It defines multiple types of transcendental Galois extensions, analyzes their properties, and establishes that purely transcendental extensions are strong Galois, expanding the understanding of Galois theory beyond algebraic cases.

Findings

Different transcendental Galois extensions are generally distinct but coincide in algebraic cases.

Properties of Galois extensions are derived using conjugation and quasi-Galois concepts.

Purely transcendental extensions are shown to be strong Galois.

Abstract

In this paper the transcendental Galois extensions of a field will be introduced as counterparts to algebraic Galois ones. There exist several types of transcendental Galois extensions of a given field, from the weakest one to the strongest one, such as Galois, tame Galois, strong Galois, and absolute Galois. The four Galois extensions are distinct from each other in general but coincide with each other for cases of algebraic extensions. The transcendental Galois extensions arise from higher relative dimensional Galois covers of arithmetic schemes. In the paper we will obtain several properties of Galois extensions in virtue of conjugation and quasi-galois and will draw a comparison between the Galois extensions. Strong Galois is more accessible. It will be proved that a purely transcendental extension is strong Galois.