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

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