Extended genus field of cyclic Kummer extensions of rational function fields

TL;DR

This paper explicitly describes the extended genus field of cyclic Kummer extensions of rational function fields, using class field theory and cohomology to analyze ambiguous classes and prime decomposition.

Contribution

It provides an explicit description of the extended genus field for cyclic Kummer extensions and establishes a reciprocity law and prime decomposition criteria.

Findings

Explicit formula for the extended genus field $K_g^+$

Cohomological determination of ambiguous class numbers

Necessary and sufficient conditions for prime decomposition

Abstract

For a cyclic Kummer extension $K$ of a rational function field $k$ is considered, via class field theory, the extended Hilbert class field $K_H^+$ of $K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along the lines of the definitions of R. Clement for such extensions of prime degree. We obtain $K_g^+$ explicitly. Also, we use cohomology to determine the number of ambiguous classes and obtain a reciprocity law for $K/k$. Finally, we present a necessary and sufficient condition for a prime of $K$ to decompose fully in $K_g^+$.