Computing generators of free modules over orders in group algebras II

TL;DR

This paper extends an algorithm for determining free modules over orders in group algebras, making it applicable in more general algebraic settings by relaxing previous restrictions.

Contribution

It generalizes a previous algorithm to work under weaker conditions on the Wedderburn components of the group algebra.

Findings

Algorithm successfully computes bases or determines non-existence in broader cases.

Generalization improves applicability to more complex algebraic structures.

Provides a practical computational tool for algebraic number theory applications.

Abstract

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. In a previous article, we gave a necessary and sufficient condition for X to be free of given rank d over A. In the case that (i) the Wedderburn decomposition of E[G] is explicitly computable and (ii) each component is in fact a matrix ring over a field, this led to an algorithm that either gives elements that either gives an A-basis for X or determines that no such basis exists. In the present article, we generalise the algorithm by weakening condition (ii) considerably.