Generators of maximal orders

TL;DR

This paper determines the minimal number of generators needed for maximal orders in semisimple algebras over number fields, extending previous results and introducing new counting techniques for finitely generated algebras.

Contribution

It provides a general method to compute the minimal number of generators for maximal orders in semisimple algebras, broadening prior work limited to number fields.

Findings

Computed minimal number of generators for maximal orders in semisimple algebras.

Extended Pleasants' results from number fields to more general algebras.

Developed new counting results for finitely generated algebras.

Abstract

Let R be the ring of algebraic integers in a number field K and let L be a maximal order in a semisimple K-algebra B. Building on our previous work, we compute the smallest number of algebra generators of L considered as an R-algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.