An elemental Erd\H{o}s-Kac theorem for algebraic number fields

Fix a number field $K$. For each nonzero $\alpha \in {\mathbb{Z}}_K$, let $\nu (\alpha )$ denote the number of distinct, nonassociate irreducible divisors of $\alpha$. We show that $\nu (\alpha )$ is normally distributed with mean proportional to $(\log \log \mathrm{\mid }N(\alpha )\mathrm{\mid }{)}^D$ and standard deviation proportional to $(\log \log \mathrm{\mid }N(\alpha )\mathrm{\mid }{)}^{D-1\mathrm{/}2}$. Here $D$, as well as the constants of proportionality, depend only on the class group of $K$. For example, for each fixed real $\lambda$, the proportion of $\alpha \in $\mathbb{Z}$[\sqrt{-5}]$with$$ \nu(\alpha) \le \frac{1}{8}(\log\log{N(\alpha)})^2 + \frac{\lambda}{2\sqrt{2}} (\log\log{N(\alpha)})^{3/2} $$isgivenby$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\lambda} e^{-t^2/2}, $\mathrm{d}$t$. As further evidence that “irreducibles play a game of chance”, we show that the values $\nu (\alpha )$ are equidistributed modulo $m$ for every fixed $m$.