Minimal Ramification Problem

Cynthia Ramiharimanana — Clemson University

Minimal ramification problem can be considered as a quantitative version of Inverse Galois Problem. The latter is a question that asks whether every finite group is a Galois group of a Galois extension of the field of rational numbers Q. It is one of the important problems in Field Arithmetic and Algebraic Number Theory. For a given finite group G, let Ram(G) be the minimal integer for which there exists a Galois extension of N/Q that ramifies exactly at Ram(G) primes. The task in minimal ramification problem is to compute or to bound Ram(G). In this talk, I will give an overview of the problem and some results, and then we will talk about current and future research.