We consider conjugacy of integral matrices by elements in $\text{ GL}_{ n}(R)$ for certain rings $R$ with subring $\mathbb{ Z}$. We note that a Hasse principal does not hold in the context of matrix conjugacy because matrices which are $\text{ GL}_{ n}(\mathbb{ Z}_p)$-conjugate for all $p$ are not necessarily $\text{ GL}_{ n}(\mathbb{ Z})$-conjugate. By a theorem of Guralnick, we know that integral $n \times n$ matrices are $\text{ GL}_{ n}(\mathbb{ Z}_p)$-conjugate for all primes $p$ if and only if they are conjugate by an element in $\text{ GL}_{ n}(E)$ for some algebraic integral extension $E$ of $\mathbb{ Z}$. We study the problem of finding this extension $E$. Since a result by Latimer and MacDuffee for describing $\mathbb{ Z}$-conjugacy can be generalized to the context of $R$-conjugacy for $R$ any integral domain, we can adapt an existing algorithm for $\mathbb{ Z}$-conjugacy to a new context. We also offer a method for finding $E$ which makes use of the principal ideal theorems of class field theory. We illustrate our method in several examples.
Categories: math.NT