Definition 13.7 (Eisenstein polynomial). $f (x) = x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{0} \in O_{K} [x]$ is Eisenstein if $v_{K} (a_{i}) \geq 1$ for all $i$ , and $v_{K} (a_{0}) = 1$ .