Extended genus fields of abelian extensions of rational function fields

TL;DR

This paper characterizes the extended genus fields of finite abelian extensions over rational function fields, providing explicit descriptions based on cyclotomic projections and their composites.

Contribution

It offers a new explicit description of extended genus fields for all finite abelian extensions of rational function fields, extending previous results for cyclic cases.

Findings

Extended genus field of cyclic prime power degree extensions characterized.

Extended genus fields of composite cyclotomic extensions are compositional.

Main result provides explicit formulas for general finite abelian extensions.

Abstract

In this paper we obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a composite of two cyclotomic extensions of a global rational function field is equal to the composite of their respective extended genus fields, to obtain our main result. This result is that the extended genus field of a general finite abelian extension of a global rational function field, is given explicitly in terms of the field and of the extended genus field of its "cyclotomic projection".