Computing generators of free modules over orders in group algebras

TL;DR

This paper develops a criterion and algorithm for determining when a lattice over an order in a group algebra is free of a given rank, with applications to Galois modules and algebraic integers, implemented in Magma.

Contribution

It provides a necessary and sufficient condition for lattices to be free over orders in group algebras and an explicit algorithm for constructing bases in certain cases.

Findings

Algorithm determines A-basis or non-existence of basis for lattices.

Applicable to Galois modules arising from number field extensions.

Implemented in Magma for specific cases when K=E=Q.

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. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.