Minimal ramification and the inverse Galois problem over the rational function field $\mathbb{F}_p(t)$

TL;DR

This paper investigates the inverse Galois problem over the rational function field _p(t), proposing a conjectural formula for the minimal ramification in Galois extensions and providing proofs for various cases.

Contribution

It introduces a conjectural formula for minimal ramification in Galois extensions over _p(t) and offers both theoretical and computational evidence supporting it.

Findings

Proposed a conjectural formula for minimal ramification.

Provided proofs for multiple cases of the conjecture.

Combined theoretical and computational methods to support the conjecture.

Abstract

The inverse Galois problem is concerned with finding a Galois extension of a field $K$ with given Galois group. In this paper we consider the particular case where the base field is $K=\F_p(t)$. We give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in $G$-extensions of $K$, and give theoretical and computational proofs for many cases of this conjecture.