The isomorphism problem for cyclic algebras and its application

Abstract

The isomorphism problem means to decide if two given finite-dimensional simple algebras over the same centre are isomorphic and, if so, to construct an isomorphism between them. A solution to this problem has applications in computational aspects of representation theory, algebraic geometry and Brauer group theory. The paper presents an algorithm for cyclic algebras that reduces the isomorphism problem to field theory and thus provides a solution if certain field theoretic problems including norm equations can be solved (this is satisfied over number fields). As an application, we can compute all automorphisms of any given cyclic algebra over a number field. A detailed example is provided which serves to construct an explicit example of a noncrossed product division algebra.