Computing generators of free modules over orders in group algebras

Let E be a number field and G be a finite group. Let A be any OE-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 E[G]and#8773;and#8853;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 X=Aα1and#8853;...and#8853;Aαd 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 OL, the ring of algebraic integers of L, and A to be the associated order A(E[G];OL)⊆E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q.