Euclidean Windows

TL;DR

This paper explores Euclidean properties of number fields using alternative functions beyond the norm, revealing that Euclidean minima with weighted norms can be irrational and non-isolated.

Contribution

It introduces the concept of Euclidean fields with respect to various functions and demonstrates new phenomena in Euclidean minima under weighted norms.

Findings

Euclidean minima can be irrational under weighted norms

Euclidean fields can be characterized using functions other than the norm

Euclidean minima may not be isolated in the number field context

Abstract

In this paper we study number fields which are Euclidean with respect to a function different from the absolute value of the norm. We also show that the Euclidean minimum with respect to weighted norms may be irrational and not isolated.