Conjugacy of Integral Matrices over Algebraic Extensions

TL;DR

This paper investigates the conjugacy of integral matrices over algebraic extensions, highlighting the limitations of local-global principles and proposing methods to find suitable algebraic extensions using class field theory.

Contribution

It extends the understanding of matrix conjugacy over rings, generalizes existing algorithms, and introduces a new approach to identify algebraic extensions via class field theory.

Findings

A Hasse principle does not hold for matrix conjugacy.

Matrices locally conjugate at all primes may not be globally conjugate over integers.

A method to find algebraic extensions E using class field theory is proposed and demonstrated.

Abstract

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.