I am trying to prove the following assertion:
Given a prime number $p\in \mathbb{Z}$ , let $f =\sum_{i=0}^{2n+1}{a_ix^i}\in \mathbb{Z}[x]$ which is a polynomial of odd degree. Furthermore, we assume $p\nmid a_{2n+1}, p^2\mid a_{0}, \ldots, a_n, p\mid a_{n+1}, \ldots, a_{2n}$ and $p^3\nmid a_0$.
The aim is to prove $f$ is irreducible. Obviously, the Eisenstien Criterion cannot be used rightaway. What I tried was to make a linear change of variables. But it did not simplify matters. Any hints?
Thanks.