Genus Fields of Congruence Function Fields

TL;DR

This paper characterizes the genus fields of finite separable extensions of rational congruence function fields, especially under ramification conditions, and provides a general description for global function fields.

Contribution

It introduces a method to determine the genus field of extensions of rational congruence function fields under specific ramification conditions, extending the understanding of genus fields in function field theory.

Findings

Explicit description of genus fields under ramification constraints

Extension of genus field theory to global function fields

Conditions for the structure of genus fields in function field extensions

Abstract

Let $k$ be a rational congruence function field and consider an arbitrary finite separable extension $K/k$. If for each prime in $k$ ramified in $K$ we have that at least one ramification index is not divided by the characteristic of $K$, we find the genus field $\g K$, except for constants, of the extension $K/k$. In general, we describe the genus field of a global function field.