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 $K$.
Categories: math.NT