 Denominator criterion for Newton polygon irreducibility

If a monic degree-$n$ polynomial over a discretely valued field with normalized valuation has all its roots of valuation $a/n$, where $\gcd (a,n)=1$, then it is irreducible. Each irreducible factor of degree $d$ has constant term of valuation $da/n\in \mathbb{Z}$, forcing $n\mid d$. Equivalently, a single Newton polygon segment of relatively prime horizontal and vertical lengths gives irreducibility.