# Transfinitely valued Euclidean domains have arbitrary indecomposable order type

## TL;DR

This paper characterizes the possible order types of Euclidean norms in transfinitely valued Euclidean domains, showing they are exactly the indecomposable ordinals, and constructs examples with specific properties.

## Contribution

It establishes a complete characterization of the order types of Euclidean norms in transfinitely valued Euclidean domains and constructs examples with unique norm properties.

## Key findings

- Every indecomposable ordinal corresponds to a transfinitely valued Euclidean domain.
- Any Euclidean norm must have an indecomposable order type.
- Constructs a finitely valued Euclidean domain without a multiplicative integer valued norm.

## Abstract

We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely characterize the norm complexity of Euclidean domains. Modifying this construction, we also find a finitely valued Euclidean domain with no multiplicative integer valued norm.