Function Fields

Introduction

By definition, a (univariate) function field can be written as a finite extension of a rational function field $k(x)$ for a field $k$ (commonly $k = \mathbb{Q}$ or $k = \mathbb{F}_p$). In Hecke, a function field $L$ is currently defined as being a (univariate) rational function field $k(x)$ or a finite extension thereof. In other words, the extension is defined in the the following way:

• We have $L = k(x)/(f)$, where $f \in k(x)[y]$ is an irreducible polynomial (simple extension)

We refer to $k(x)$ as the base field of the function field $L$. We call $L$ an absolute function field if the base field is equal to the rational function field $k(x)$.