An equation defined by norm

Question

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that $$N_{K/ {\mathbb{Q}}}(\gamma)\equiv a^n\ \ (mod\ p) $$