Terminating Euclidean algorithm for a non-Noetherian Bézout domain and an algorithm to find the normal forms of matrices in GL2(k[x,y])

Abstract

In this dissertation, we will define a Euclidean-like norm and a division algorithm for a non-Noetherian Bézout domain, k[y]+x k(y)[x], where k is a field. And we will show that the Euclidean algorithm for that domain always terminates. As its application, we will give an algorithm to find the normal form of any matrix in GL2(k[x,y]) over k, with respect to the amalgamated free product structure.