In Ireland and Rosen's book on number theory they give a proof of the finiteness of the class group of a number field which they attribute to Hurwitz, but which is essentially due to Kronecker (as I learnt from a comment by Franz Lemmermeyer to this answer by KConrad). The proof is based on showing that the ring of integers in a number field satisfies the following property which I here formulate for an arbitrary integral domain.

Let's call an integral domain $R$ a *Kronecker--Hurwitz domain* if there exists a function
$f:R\rightarrow\mathbb{Z}_{\geq0}$ and a finite subset $T\subset R\setminus\{0\}$
such that

- $f(\alpha)=0$ if and only if $\alpha=0$;
- for any $\alpha,\beta\in R$, $\beta\neq0$, there is a $t\in T$ and an $\omega\in R$ such that $f(t\alpha-\omega\beta)<f(\beta)$.

Perhaps it is natural to also assume that $f$ is multiplicative so that the above inequality can be written as $f(t γ - ω ) < 1$, for any $γ$ in the field of fractions of $R$ (this is what is actually proved in Ireland--Rosen when $f$ is the absolute value of the field norm).

At first sight, this looks a bit like the conditions for a Dedekind--Hasse norm, but of course there is a massive difference because if $R$ has a Dedekind--Hasse norm then it must be a PID.

**Question**. Let $R$ be a ring of integers in a global function field (i.e., $R$ is the integral closure of $\mathbb{F}_q[t]$, for some $t$ in the fraction field, transcendent over $\mathbb{F}_q$). Is $R$ a Kronecker--Hurwitz domain?

I would also be interested to know whether the defining property of Kronecker--Hurwitz domains has appeared anywhere in the literature.

I have asked a couple of experts about this, but have so far not made any progress towards an answer.

**Note.** I have not been able to adapt the proof in Ireland--Rosen (p. 178) that rings of integers in number fields are Kronecker--Hurwitz to the function field case because it is based on embedding the coordinates of an element w.r.t. an integral basis into the Euclidean space $\mathbb{R}^n$. More precisely, let $N$ be the ideal norm on $\mathbb{F}_q[t]$ extended to $\mathbb{F}_q(t)$. Partition the unit cube $\{x\in\mathbb{F}_{q}(t)\mid0\leq N(x)\leq1\}^{n}$
into $m^{n}$ subcubes with side length $1/m$. At the end of the
proof, we obtain a $\delta=\{h\gamma\}-\{l\gamma\}$ with $\{h\gamma\}$
and $\{l\gamma\}$ in the same subcube. This means that the $\mathbb{F}_{q}(t)$-coordinates
of $\{h\gamma\}_{i}$ and $\{l\gamma\}_{i}$ of $\{h\gamma\}$ and
$\{l\gamma\}$ satisfy
$$
\frac{a_{i}}{m}\leq N(\{h\gamma\}_{i}),N(\{l\gamma\}_{i})\leq\frac{a_{i}+1}{m},
$$
for some integers $0\leq a_{i}\leq m-1$. For the proof to go through,
however, we would need $N(\{h\gamma\}_{i}-\{l\gamma\}_{i})\leq1/m$,
which does not follow from the above inequalities in the function
field case (e.g., if $m=q^{2}$, $a_{1}=q-1$, $\{h\gamma\}_{1}=\frac{1}{t}$,
$\{l\gamma\}_1=\frac{2}{t}$, assuming $\text{char }\mathbb{F}_{q}>2$;
then $N(\{h\gamma\}_{1})=N(\{l\gamma\}_{1})=1/q=\frac{a_{i}+1}{m}$,
but $N(\{h\gamma\}_{1}-\{l\gamma\}_{1})=N(-1/t)=1/q>1/m$).