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[k:Q],G (X^ 4√|G|). This improves over the previous best bound Ok,G,ε(X3/8+ε) due to Ellenberg and Venkatesh. In particular, ours is the first bound for general G with an exponent that decays as |G| → ∞.