# Ramification in the Inverse Galois Problem

## TL;DR

This paper investigates the inverse Galois problem with a focus on ramification restrictions, presenting new results for extensions with limited ramification and examining a conjecture relating Galois group generators to ramified primes.

## Contribution

It provides new results on ramification in inverse Galois extensions and verifies Harbater's conjecture for certain classes of groups, advancing understanding of ramification constraints.

## Key findings

- Proved Harbater's conjecture for groups with a nilpotent subgroup of index 1, 2, or 3.
- Established results for extensions ramified at a single prime.
- Analyzed the structure of Galois groups under ramification restrictions.

## Abstract

This paper focuses on a refinement of the inverse Galois problem. We explore what finite groups appear as the Galois group of an extension of the rational numbers in which only a predetermined set of primes may ramify. After presenting new results regarding extensions in which only a single finite prime ramifies, we move on to studying the more complex situation in which multiple primes from a finite set of arbitrary size may ramify. We then continue by examining a conjecture of Harbater that the minimal number of generators of the Galois group of a tame, Galois extension of the rational numbers is bounded above by the sum of a constant and the logarithm of the product of the ramified primes. We prove the validity of Harbater's conjecture in a number of cases, including the situation where we restrict our attention to finite groups containing a nilpotent subgroup of index $1,2,$ or $3$. We also derive some consequences that are implied by the truth of this conjecture.