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Galois Seminar

Friday, October 20, 2017 - 3:15pm

Ben Pollak

University of Pennsylvania


University of Pennsylvania

DRL 4N30

In addition to simply asking whether every finite group is the Galois group of a Galois extension of the rationals, it is of interest to study the ramification in the extensions that realize a given group. In this talk I will explore the relationship between sets of integral primes, S, and the finite groups appearing as Galois groups of extensions of the rationals unramified outside of S. I will start by reviewing the situation for abelian groups, and then move on to nilpotent, solvable, and non-solvable groups. In each case, I will remark on a conjecture due to David Harbater that relates the product of the ramified primes in a finite, Galois extension of the rationals to the smallest size of any generating set of its Galois group, as well as a conjecture due to Nigel Boston and Nadya Markin on the minimal number of ramified primes needed to realize a given finite group as the Galois group of an extension of the rationals.