Let $p \in \mathbb{Z}$ be a prime, and let $f(x) = px^n + \dots$ be an irreducible degree $n$ polynomial over $\mathbb{Z}$ with leading coefficient equal to $p$. Suppose that $f(x)$ has no repeated roots modulo $p$. Then $p$ is unramified in the number field $K = \mathbb{Q}[x]/(f(x))$ and splits into distinct prime ideals $(p) = \prod_{i = 1}^r \mathfrak{p}_i$ of ramification degree $1$.
Question: Let $\theta$ be the image of $x$ in $K$. Can anything be said about the multiplicity of each prime ideal $\mathfrak{p}_i$ in the prime factorization of $\theta$?
What I know: If $p^a$ divides the constant term of $f$, then $p^{a-1} \mid \operatorname{Norm}(\theta)$. Also, since $p \theta$ is an algebraic integer, we know that every prime factor of $p$ appearing with negative multiplicity in $\theta$ must appear with multiplicity $-1$, but I don't know what else can be deduced.