Help with proof involving Eisenstein's criterion Let $a\in\mathbb {Z}[X]$ and suppose that $2a\in\mathbb {Z}[X]$ is Eisenstein with respect to a prime $p\in\mathbb {Z}$. 
How can I prove that $a$ is an Eisenstein polynomial with respect to $p$?
Any help would be greatly appreciated!

Hint 1 Show that $p \neq 2$. To do this, use $p\nmid 2a_n$.
Hint 2: If $p|2a_k$ and $p \neq 2$ deduce that $p|a_k$.
Hint 3: If $p^2 \nmid 2a_0$ show that $p^2 \nmid a_0$. 
