Unramified Extension of a $p$-Adic Field

Let $K/k$ be a finite extension of nonarchimedean local fields (e.g. $p$-adic fields).

The extension is unramified if its ramification index $e(K/k)=1$ (equivalently, $\mathfrak p_k\mathcal O_K=\mathfrak p_K$).

Equivalently, $\mathcal O_K/\mathfrak p_K$ is a finite extension of $\mathcal O_k/\mathfrak p_k$ of degree $[K:k]$.

Key fact: in the Galois unramified case, there is a canonical Frobenius element generating $\mathrm{Gal}(K/k)$.