An Eisenstein-like irreducibility criterion

Solution 1:

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

Solution 2:

One way to prove the irreducibility seems to be to use the Newton Polygon. The condition on the coefficients of $f$ means that the Newton Polygon has a side of slope $\dfrac{1}{n}-1$ and hence that $f$ has a root $\alpha$ in some algebraic closure $F$ of $\mathbf{Q}_p$ having valuation $1 - \dfrac{1}{n}$ (there is a unique way to prolong $e_p$ to a valuation of $F$).

But then the extension $\mathbf{Q}_p \subset \mathbf{Q}_p(\alpha)$ is totally ramified of degree $n$ and $f$ must be irreducible over $\mathbf{Q}_p$ hence a fortiori irreducible over $\mathbf{Q}$.