Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of degree $n$, and let $K$ be the splitting field of $f$ over $\mathbb{Q}$. Consider the ring of integers $\mathcal{O}_K$ of the Galois number field $K$.
Take a prime number $p\in\mathbb{Z}$. I'm interested in an upper bound, in terms of $n$, on the number of distinct prime ideals of $\mathcal{O}_K$ that lie above $p$.
An obvious upper bound is $n!$ because this is an upper bound on $[K:\mathbb{Q}]$. Is there a better upper bound? Is there an upper bound only exponential in $n$, or even polynomial in $n$?