Variants of Eisenstein irreducibility

Basically all such criteria boil down to some argument involving the Newton polygon as Kevin Buzzard mentions in the comments. While something as general as your statement has trivial counter examples the following generalization holds:

Let $R$ be a unique factorization domain and $f(x) =a_nx^n+\cdots +a_0\in R[x]$ with $a_0a_n\neq 0$. If the Newton polygon of $f$ with respect to some prime $p\in R$ consists of the only line segment from $(0,m)$ to $(n, 0)$ and if $gcd(n,m) = 1$ then $f$ is irreducible in $R[X]$.

I've heard this called the Eisenstein-Dumas criterion of irreducibility (it also proves the example given in the comments). Another generalization of Eisenstein's criterion is the following:

If $p|a_0,a_1,\dots,a_k$ but $p^2\not | a_0$ then $f(x)$ has an irreducible factor of degree $\geq k+1$

(This is how you prove for example, that a polynomial like $x^n+5x^{n-1}+3$ is irreducible, after checking that it has no linear factors.) If not answering your question, at least I hope that this refreshes your memory of the statement you claim above. :)

Such a generalization (Dumas' theorem) was discussed here: Is a polynomial with 1 very large coefficient irreducible?

A good source to learn about it is Prasolov's book on polynomials: http://tinyurl.com/prasolov - see page 53, Dumas' theorem (and a bit before this theorem).

I've meanwhile found something in

• S. MacLane, The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals, Trans. Amer. math. Soc. 43 (1938), 226--239.

MacLane referred, among others, to the article

• E. Netto, Ueber die Irreductibilität ganzzahliger ganzer Functionen, Math. Ann. 48 (1897), 81--88

There, Netto proved the following: A polynomial $$ f(x) = x^n + a_{n-1} p x^{n-1} + \ldots + a_{k+1} p z^{k+1} + a_k p^2 z^k + \ldots + a_0 p^2 $$ with degree $n > 2k$, in which the $a_j$ are integers such that $p \nmid a_0$, does not have a factor of degree less than $k+1$. This is similar to Gjergji's second example, but allows the divisibility by $p^2$ that I had had in mind.