Euclidean domains with no multiplicative norms

TL;DR

This paper constructs a Euclidean domain lacking a multiplicative Euclidean norm to a well-ordered monoid, demonstrating that such domains can exist without multiplicative norms to the real numbers, and explores the preservation of UFD properties during certain extensions.

Contribution

It introduces a novel Euclidean domain without a multiplicative Euclidean norm to a well-ordered monoid, expanding understanding of Euclidean domains and their properties.

Findings

Constructed a Euclidean domain with no multiplicative Euclidean norm.

Proved the UFD property is preserved when adjoining a free factorization.

Showed the non-existence of a multiplicative Euclidean norm to c9 under the usual order.

Abstract

We construct a Euclidean domain with no multiplicative Euclidean norm to a compatibly well-ordered monoid, and hence with no multiplicative Euclidean norm to $\mathbb{R}$ (under its usual order). A key step in the proof is showing that the UFD property is preserved when adjoining a free factorization.