On the computation of overorders

TL;DR

This paper develops efficient algorithms for computing all overorders and minimal overorders of a given order in a semisimple algebra, advancing computational methods in algebraic number theory.

Contribution

It introduces practical algorithms for overorder computations using representation theory and minimal ring extension theory, with publicly available implementations.

Findings

Algorithms successfully compute all overorders and minimal overorders

Implementation demonstrates practical efficiency

Advances computational techniques in algebraic number theory

Abstract

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper we consider the related problems of computing all minimal overorders as well as all overorders of a given order. We use techniques from algorithmic representation theory and the theory of minimal integral ring extensions to obtain efficient and practical algorithms, whose implementation is publicly available.