TITLE: irreducible polynomials over the $p$ adic number
QUESTION [6 upvotes]: Are there some necessary conditions on the coefficients of a monic, irreducible polynomial $\Phi$ over a local field?

REPLY [7 votes]: A corollary of Hensels lemma: 
If K is a field that is complete with respect to a non-archimedean absolute value, then for an irreducible polynomial $P=\sum{a_iX^i}$ of degree $n$ we have 
max($|a_i|$)=max($|a_0|$,$|a_n|$)