Computing generators of free modules over orders in group algebras

Let E be a number field and G be a finite group. Let be any -order of full rank in the group algebra E[G] and X be a (left) -lattice. We give a necessary and sufficient condition for X to be free of given rank d over . In the case that the Wedderburn decomposition E[G] χMχ is explicitly computable and each Mχ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α1,…,αd X such that or determines that no such elements exist. 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 , the ring of algebraic integers of L, and to be the associated order . The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when .