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Euclidean domains with no multiplicative norms

Caleb J. Dastrup, Pace P. Nielsen

TL;DR

The paper constructs a concrete Euclidean domain R that admits no multiplicative Euclidean norm to any compatibly well-ordered monoid, and hence no multiplicative norm into the usual ordered real numbers. The core method is a transfinite, iterative construction that freely adjoins prime factorizations while preserving unique factorization, culminating in R_infty and a localization R in which the Euclidean division can be forced to fail for multiplicative norms. A key technical ingredient is a rank-based analysis showing that certain remainders necessarily split into at least two primes, contradicting multiplicative monotonicity arguments, while an explicit N-valued norm is still defined to demonstrate R remains Euclidean. The paper also extends the construction to rule out broader monotonicity properties and discusses implications for the existence of multiplicative norms in Euclidean domains, highlighting fundamental limitations in transferring Euclidean structure to multiplicative norm frameworks.

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.

Euclidean domains with no multiplicative norms

TL;DR

The paper constructs a concrete Euclidean domain R that admits no multiplicative Euclidean norm to any compatibly well-ordered monoid, and hence no multiplicative norm into the usual ordered real numbers. The core method is a transfinite, iterative construction that freely adjoins prime factorizations while preserving unique factorization, culminating in R_infty and a localization R in which the Euclidean division can be forced to fail for multiplicative norms. A key technical ingredient is a rank-based analysis showing that certain remainders necessarily split into at least two primes, contradicting multiplicative monotonicity arguments, while an explicit N-valued norm is still defined to demonstrate R remains Euclidean. The paper also extends the construction to rule out broader monotonicity properties and discusses implications for the existence of multiplicative norms in Euclidean domains, highlighting fundamental limitations in transferring Euclidean structure to multiplicative norm frameworks.

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 (under its usual order). A key step in the proof is showing that the UFD property is preserved when adjoining a free factorization.
Paper Structure (5 sections, 15 theorems, 39 equations)

This paper contains 5 sections, 15 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. UFDs and splitting primes
  3. Main construction
  4. Monotonicity considerations
  5. Acknowledgements

Key Result

Proposition 1.2

Let $R$ be a Euclidean domain. There is a well-ordering $\prec$ on $R-\{0\}$ such that with respect to this ordering the identity map on $R-\{0\}$ is a (possibly transfinite) Euclidean norm for $R$.

Theorems & Definitions (30)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 20 more