Computing generators of free modules over orders in group algebras

Journal article

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]and#8773;and#8853;_χMχ 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 α₁,…,α_d∈X such that X=Aα₁and#8853;^...and#8853;Aα_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.

Authors: Bley, W; Johnston, H

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: JOURNAL OF ALGEBRA
• Volume: 320
• Issue: 2
• Pages: 836-852
• Publication date: 2008-07-15
• DOI: 10.1016/j.jalgebra.2008.01.042
• ISSN: 0021-8693
• Language: English
• Keywords: Galois module structure; associated orders; group algebras