  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.

Rings and Algebras