Computing generators of free modules over orders in group algebras

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 E [ G ] ≅ ⊕ χ M χ is explicitly computable and each M χ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α 1 , … , α d ∈ X such that X = A α 1 ⊕ ⋯ ⊕ A α d 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 O L , the ring of algebraic integers of L, and A to be the associated order A ( E [ G ] ; O L ) ⊆ E [ G ] . The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = Q .