# Multiplicative Euclidean Function for an Euclidean Domain

Does there exist an Euclidean domain with no multiplicative Euclidean function?

An Euclidean domain, denoted $R$, is an integral domain

with an Euclidean function $d : R\setminus \{0\} \to \mathbb{N}$ such that

$1)\quad d(a) \leq d(ab)$, and

$2)\quad a = bq + r$ with either $r = 0$ or $d(r) < d(b)$.

I am interested in multiplicative Euclidean functions. That is, $d(ab) = d(a)d(b)$.

For example, one can choose

$\mathbb{Z}$ with $d(n) = |n|, \quad \mathbb{Z}[i]$ with $d(\alpha) = N(\alpha), \quad F[X]$ with $d(f) = 2^{\deg(f)}$ for a field $F$,

or, kind of a stupid example, any field $F$ with $d(a) = 1$.

I am new here. Thank you in advance.

$$d_{\min}(a) = \min\, \{d(a)\ :\ d\ \text{ is a Euclidean function on}\ R\}$$
This question has now been answered in the negative. There is a Euclidean domain but no Euclidean function for that ring into $\mathbb{N}$ is multiplicative. See my paper here.