Eisenstein polynomial

Last updated January 27, 2022

Definition

Let

• $\mathcal{O}$ be a complete discrete valuation ring
• $\mathfrak{p}$ its maximal ideal

A separable polyomial $$P(T)=T^n+\sum_{i=0}^{n-1}a_iT^i\in\mathcal{O}[T]$$ is called Eisenstein iff all $a_i\in\mathfrak{p}$ and $v_\mathfrak{p}(a_0)=1$.

Properties

1. An Eisenstein polynomial is irreducible