Enumerating Galois extensions of number fields

We provide an asymptotic formula for the number of Galois extensions of a number field with absolute discriminant bounded by some X. The key behind this result is a new upper bound on the number of Galois extensions with a given Galois group G of order at least 5. In particular, we give the first general bound with an exponent that decays with the order of G. This improves over the previous best bound due to Ellenberg and Venkatesh.

Robert has also kindly agreed to give a pretalk on upper bounds on number fields from 2:00-2:45pm in SC 530. We especially encourage younger students to attend and ask questions!