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

TL;DR

This paper investigates the asymptotic distribution of extensions of central simple algebras over number fields, extending classical field extension results to noncommutative algebraic structures.

Contribution

It provides asymptotic formulas for counting Galois extensions of fixed degree and relates the distribution of outer extensions to field extensions of the center.

Findings

Asymptotic counts for inner Galois extensions with bounded discriminant

Relations between outer extensions of algebras and extensions of their centers

Generalization of field extension asymptotics to noncommutative 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.