Asymptotics of extensions of simple $\mathbb{Q}$-algebras

Abstract

We answer various questions concerning the distribution of extensions of a given central simple algebra $K$ over a number field. Specifically, we give asymptotics for the count of inner Galois extensions $L/K$ of fixed degree and center with bounded discriminant. We also relate the distribution of outer extensions of $K$ to the distribution of field extensions of its center $Z(K)$. This paper generalizes the study of asymptotics of field extensions to the noncommutative case in an analogous manner to the program initiated by Deschamps and Legrand to extend inverse Galois theory to division algebras.

33 pages. Final version. To appear in Algebra & Number Theory