Similarity of Matrices over Dedekind Rings

TL;DR

This paper generalizes a classical theorem to broader algebraic settings and investigates matrix similarity over rings of algebraic integers, proposing a conjecture and verifying it in specific cases.

Contribution

It extends Latimer and MacDuffee's theorem to general commutative domains and explores matrix similarity over integral rings of number fields, including a conjecture on descent over valuation rings.

Findings

Extended theorem to commutative domains

Verified conjecture for 2x2 matrices

Provided insights into similarity over algebraic integer rings

Abstract

We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent by a finite covering and verify this conjecture for $2\times2$ matrices and separable characteristic polynomials.