  Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two
asymptotic formulas connected with the distribution of irreducible elements in
$\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated
irreducibles dividing a nonzero element of $\mathbb{Z}_K$ of norm not exceeding
$x$ (in absolute value), as $x\to\infty$. Second, we count the number of
irreducible elements of $\mathbb{Z}_K$ of norm not exceeding $x$ lying in a
given arithmetic progression (again, as $x\to\infty$). When $K=\mathbb{Q}$,
both results are classical; a new feature in the general case is the influence
of combinatorial properties of the class group of $K$.
