Enumerating Galois extensions of number fields

Let k be a number field. We provide an asymptotic formula for the number of Galois extensions of k with absolute discriminant bounded by some X 1, as X. We also provide an asymptotic formula for the closely related count of extensions K/k whose normal closure has discriminant bounded by X. The key behind these results is a new upper bound on the number of Galois extensions of k with a given Galois group G and discriminant bounded by X; we show the number of such extensions is O₊: ₐ, ₆ (X^ 4{|G|}). This improves over the previous best bound O₊, ₆, (X^3{8+}) due to Ellenberg and Venkatesh. In particular, ours is the first bound for general G with an exponent that decays as |G|.