Asymptotics of extensions of simple ℚ-algebras

Gundlach, Fabian; Seguin, Béranger

doi:10.2140/ant.2026.20.383

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

Fabian Gundlach and Béranger Seguin

Vol. 20 (2026), No. 2, 383–418

DOI: 10.2140/ant.2026.20.383

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.

Keywords

simple algebras, Malle's conjecture, counting problems

Mathematical Subject Classification

Primary: 11N45, 12E15, 11R52

Milestones

Received: 28 June 2024

Revised: 17 December 2024

Accepted: 20 January 2025

Published: 16 February 2026

Authors

Fabian Gundlach
Fakultät EIM, Institut für Mathematik Universität Paderborn 33098 Paderborn Germany

Béranger Seguin
Fakultät EIM, Institut für Mathematik Universität Paderborn 33098 Paderborn Germany

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Vol. 20, No. 2, 2026