Eisenstein polynomial (reviewed)

Let $D$ be an integral domain and $P$ a prime ideal in $D$. A polynomial $a_n x^n + \cdots + a_0\in D[x]$ is Eisenstein (with respect to $P$) if

• $a_n \not\in P$
• $a_i \in P$ for all $0\leq i < n$
• $a_0 \not\in P^2$

An Eisenstein polynomial is irreducible in $F[x]$ where $F$ is the field of fractions of $D$.

If $K$ is a finite extension of $\Q_p$ with ring of integers $\mathcal{O}_K$, then a finite extension $L/K$ is totally ramified if and only if $L=K(\alpha)$ where $\alpha$ is a root of an Eisenstein polynomial in $\mathcal{O}_K[x]$.