Looking for more direct proof of non-existence of "Euclidean algorithm norm" for $\mathbb{Z}[\sqrt{-5}]$

Question

The textbook I'm reading, Integers, Polynomials, and Rings, by Ronald Irving, states on p. 246:

A ring $R$ is called a Euclidean ring if it satisfies the following three properties:

A. There is a norm function $N$ assigning to every nonzero element $a$ of $R$ a nonnegative integer $N(a)$ and assigning to $0$ a value $N(0)$ less than the norm of every nonzero element of $R$.

B. For any two nonzero elements $a$ and $b$ of $R$, $$N(a) \le N(ab)\;.$$

C. For any two nonzero elements $a$ and $b$ of $R$, there exist elements $q$ and $r$ such that $$b = aq + r$$ and $N(r) < N(a)$.

After this, the book builds a tower of about a half dozen theorems culminating in the unique factorization theorem for Euclidean spaces.

Now, in the ring $\mathbb{Z}[\sqrt{-5}]$, the elements $2$, $3$, $1 + \sqrt{-5}$, and $1 - \sqrt{-5}$ are irreducible, but they produce distinct factorizations of the element $6$: $$2\cdot 3 = (1 + \sqrt{-5})\cdot (1 - \sqrt{-5})\;.$$ This implies that no norm $N$ can exist for $\mathbb{Z}[\sqrt{-5}]$ that satisfies the three properties listed above.

Is there a more direct proof of the nonexistence of such norm for $\mathbb{Z}[\sqrt{-5}]$ than such a violation of unique factorization?

Answer 1

The question is what is meant by "a more direct proof". More direct does not mean easier. And it is certainly easier to refute UFD or PID instead of directly refuting norm-Euclidean. For example, it is easy to see that this ring is not a PID:

To show $\mathbb{Z}[\sqrt{-5}]$ is not a Euclidean domain, why suffices to show only the field norm $N(a+b\sqrt{-5})=a^2+5b^2$ doesn't work?

Answer 2

What you call norm function in your post is usually referred to as euclidian function or stathm. I think the official terminology is better, in order to avoid confusion with the norm map in ANT, the point being that there are domains which are euclidian w.r.t. a stathm which does not come from the norm. See e.g. Lemmermeyer's survey www.rzuser.uni-heidelberg.de/~hb3/publ/survey.pdf.

As stressed by Dietrich Burde, for a given domain $R$, it could be more difficult to give a direct characterisation than an indirect one. But a unified characterisation, however complicated, could produce a general answer for a family of domains. Example: the Motzkin sets in an integral domain $R$ are defined recursively by $E_0= (0), E_1=(0) \cup R^*, E_k = (0) \cup $ {$a\in R$ s.t. each residue class mod $a$ contains an element $b\in E_{k-1}$ }, $E_{\infty}$ = union of all the $E_k$ 's, and Motzkin's criterion states that $R$ is an euclidian domain iff $R=E_{\infty}$. With a bonus: the associated stathm $f$ is defined by $f(r)$ = the least index $k$ s.t. $r\in E_k$. This can be used to determine all the imaginary quadratic fields $\mathbf Q(\sqrt -d)$ whose rings of integers are euclidian. The list is: d=1, 2, 3, 7, 11, and the corresponding stathm is the norm in all cases (historically, this came as the completion of the uncoordinate efforts of many people using dissimilar methods, see Lemmermeyer, op. cit.).