A partial solution to the similarity extension problem

Abstract

The Similarity Extension Theorem roughly states that two matrices are similar over every localisation of a specific ring if and only if there is a finite integral extension of that ring where they are similar. Since the Similarity Extension Theorem was proved a question that remained was finding this extension. The goal of this thesis is to dive into this problem and propose a method of solving it in a specific setting, with the hope that this can be extended to a broader setting in the future. This was done by going over the proof of the Similarity Extension Theorem and trying to adapt it by using a theorem from Watson such that it returns an explicit extension. We succeeded in finding a method that returns a solution for two 2x2 matrices with some additional conditions. However we suspect some of these conditions can be avoided or relaxed with some additional research. This method therefore could be a good addition to current methods whenever we are looking at 2x2 matrices. And can be a start for anyone trying to generalise this method to larger matrices.