## A bound on the number of irreducible factors of an element in an algebraic integer ring

Let $\alpha$ be an algebraic integer in an algebraic number field $K$. Let $N$ denote the norm over $K$. Prove that the number of factors in any irreducible factorization of $\alpha$ over $K$ is at most $\log_2|N(\alpha)|$.

Note that $\log_2|N(\alpha)|$ is some real number between $t$ and $t+1$, where $2^t \leq |N(\alpha)| < 2^{t+1}$. No natural number less than $2^{t+1}$ can have $t+1$ prime factors (including multiplicity), and since $N$ is multiplicative, the number of irreducible factors of $\alpha$ is bounded above by the number of prime factors of $N(\alpha)$. So the number of irreducible factors of $\alpha$ in $K$ is at most $\log_2|N(\alpha)|$.