Exercise 20, § 3

Let $A$ be a factorial ring, $f = \sum_{k=0}^n a_k X^k$ a polynomial of $A[X]$; suppose that there exists an extremal element $p$ of $A$ such that : 1°) there exists an index $k \leq n$ such that $a_k$ is not divisible by $p$, but $a_i$ is divisible by $p$ for $i < k$; 2°) $a_0$ is divisible by $p$, but not by $p^2$. Now prove that under these conditions, one of the irreducible factors of $f$ in $A[X]$ is of degree $\geq k$ (reason in $(A/pA)[X]$). Special case where $k = n$ (« Eisenstein irreducibility criterion »).