Ramification in the Inverse Galois Problem
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 or . We also derive some consequences that are implied by the truth of this conjecture.
Cite
@article{arxiv.1905.01363,
title = {Ramification in the Inverse Galois Problem},
author = {Benjamin Pollak},
journal= {arXiv preprint arXiv:1905.01363},
year = {2019}
}