Denominator criterion for Newton polygon irreducibility
= 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.