Kummer theory for function fields

TL;DR

This paper extends Kummer theory to algebraic function fields with multiple variables, analyzing Galois groups of finitely generated Kummer extensions and linking their properties to constant fields.

Contribution

It develops a framework for understanding Kummer extensions over algebraic function fields, including those over cyclotomic extensions, and relates their Galois groups to constant field extensions.

Findings

Galois group structures are characterized for finitely generated Kummer extensions.

Degree computations of extensions reduce to constant field cases.

Results unify Kummer theory for function fields with classical cases.

Abstract

We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the structure of its Galois group. Our results show in a precise sense how the questions of computing the degrees of these extensions and of computing the group structures of their Galois groups reduce to the corresponding questions for the Kummer extensions of their constant fields.