Definition 2

Let $A$ be a ring, $p$ a prime ideal of $A$, and $P$ a polynomial in $A[T]$. One says that $P$ is an Eisenstein polynomial for $p$ if it satisfies the following conditions: a) $P$ is monic of degree $d \geqslant 1$; b) one has $P(T) \equiv T^d \mod. pA[T]$; c) one has $P(0) \notin p^2$.

In other words, a polynomial of Eisenstein for $p$ is a polynomial of the form $P(T) = T^d + \sum_{i=1}^d a_i T^{d-i}$, with $d \geqslant 1$, where $a_1, ..., a_{d-1}$ belong to $p$ and $a_d$ to $p - p^2$.

One says that $P$ is a polynomial of Eisenstein for $pA_p$ if the canonical image of $P$ in the polynomial ring $A_p[T]$ is a polynomial of Eisenstein for the ideal $pA_p$; this also means that $P$ is a polynomial of Eisenstein for $p$ and that it moreover satisfies the following condition, stronger than c): c’) every element $a$ of $A$ such that $aP(0) \in p^2$ belongs to $p$.