Small generators of function fields

TL;DR

This paper proves the existence of small generators for finite extensions of function fields over global fields, addressing a question related to minimal generating elements in number fields.

Contribution

It establishes the existence of small generators in the function field setting, extending Ruppert's question from number fields to function fields.

Findings

Existence of small generators for function field extensions.

Extension of Ruppert's question to the function field case.

Provides a constructive approach to find small generators.

Abstract

Let $K/k$ be a finite extension of a global field. Such an extension can be generated over $k$ by a single element. The aim of this article is to prove the existence of a "small" generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.