The minimal ramification problem for rational function fields over finite fields

TL;DR

This paper investigates the minimal ramification in Galois extensions of rational function fields over finite fields, proposing a conjecture and establishing results for specific groups.

Contribution

It introduces a new conjecture on minimal ramification in function fields and proves it for abelian, symmetric, and alternating groups in many cases.

Findings

Conjecture on minimal ramification for rational function fields.

Proved cases for abelian, symmetric, and alternating groups.

Provides a framework analogous to number field cases.

Abstract

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.