In this dissertation, we consider R-conjugacy of integral matrices for various commutative rings R. An existence theorem of Guralnick states that integral matrices which are Zp-conjugate for every prime p are conjugate over some algebraic extension of Z. We refer to the problem of determining this algebraic extension as the conjugacy extension problem. We will describe our contributions to solving this problem. We discuss how a correspondence between Z-conjugacy classes of matrices and certain fractional ideal classes can be extended to the context of R-conjugacy for R an integral domain. In the case of integral matrices with a fixed irreducible characteristic polynomial, this theory allows us to implement an algorithm which tests for conjugacy of these matrices over the ring of integers of a specified number field. We also describe how class fields can be used to solve the conjugacy extension problem in some examples.
