Euclidean rings of affine curves (Q2277523)

This paper completes the classification of algebras of finite type over a field which are Euclidean domains, begun by Armitage, Samuel, Leitzel, Madan, Queen, Lenstra, and the author. The main point proved here, left open by the previous work, is that Euclidean algebras of finite type over infinite ground fields are coordinate rings of affine open subschemes of the projective line \({\mathbb{P}}^ 1\). - A classification of Euclidean rings of S-integers in algebraic number fields was given, conditionally upon the generalised Riemann hypothesis, by Cooke, Weinberger, and Lenstra. Several consequences of the classification are given; in particular, a question of Samuel is answered affirmatively: namely, that a principal ideal domain, which is an algebra of finite type over a field, is Euclidean if and only if the (so-called) minimal algorithm does not terminate at the second stage. - The proof of the main result uses Diophantine geometry. More precisely, it is based on Siegel's finiteness theorem for integral points on curves over number fields and an analogue proved in this paper for curves over fields of positive characteristic.