show · lf.eisenstein_polynomial all knowls · up · search:

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 $\mathcal{O}_K[x]$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2018-08-06 15:11:41
Referred to by:
History: (expand/hide all)