Euclidean rings of $S$-integers in complex quadratic fields

TL;DR

This paper investigates whether rings of $S$-integers in complex quadratic fields are Euclidean domains using an elementary approach focused on the $S$-norm, contributing to the understanding of algebraic number theory structures.

Contribution

It provides a new elementary method to determine Euclideanity of $S$-integer rings in complex quadratic fields, enhancing previous approaches.

Findings

Identifies conditions under which $S$-integer rings are Euclidean

Develops an elementary criterion based on the $S$-norm

Simplifies the analysis of Euclideanity in complex quadratic fields

Abstract

We give an elementary approach to studying whether rings of $S$-integers in complex quadratic fields are Euclidean with respect to the $S$-norm.