An Eisenstein-like irreducibility criterion I could use some help with proving the following irreducibility criterion. (It came up in class and got me interested.)
Let p be a prime. For an integer $n = p^k n_0$, where p doesn't divide $n_0$, set: $e_p(n) = k$. Let $f(x) = a_n x^n + \cdots + a_1 x + a_0$ be a polynomial with integer coefficients. If:


*

*$e_p(a_n) = 0$, 

*$e_p(a_i) \geq n - i$, where $i = 1, 2, \ldots, n-1$,

*$e_p(a_0) = n - 1$,


then f is irreducible over the rationals. 
Reducing mod p and mimicking the proof of Eisenstein's criterion doesn't cut it (I think). I also tried playing with reduction mod $p^k$, but got stuck since $Z_{p^k}[X]$ is not a UFD.
Also, does this criterion has a name?

Apply Eisenstein's criterion to ${1 \over p^{n-1}}x^nf({p \over x})$.
