Exercise 20, § 3

Let A be a factorial domain and $f = \sum_{k=0}^n a_k X^k$ a polynomial in $A[X]$: suppose that there exists an extremal element $p$ in A such that: (1) there exists an index $k \leq n$ such that $a_i$ is not divisible by $p$ but $a_i$ is divisible by $p$ for $i < k$; (2) $a_i$ is divisible by $p$ but not by $p^2$. Show that under these conditions one of the irreducible factors off in $A[X]$ is of degree $\geq k$ (argue in $(A/pA)[X]$). Consider the particular case where $k = n$ ("Eisenstein's irreducibility criterion").