In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he claimed that the result also holds in the following case: assume that $$ f(x) = x^m + a_{m-1}x^{m-1} + \ldots + a_0, $$ where the $a_i$ are integers divisible by $p$, and where $p^2 \nmid a_j$ for some $0 \le j \le m-1$. There is an obvious counterexample provided by the quadratic polynomial $f(x) = (x-p)^2 = x^2 -2px + p^2$, so that could be the end of that story. But it isn't: I've read somewhere that some form of this criterion holds for polynomials of degree $\ge 3$, and that the degrees of the possible factors of counterexamples can be predicted in terms of this index $j$. Unfortunately, I don't remember the exact statement (those who are familiar with Newton polygons will probably be able to figure out a correct version) or where I've seen this. Can anyone help?