Newton polygon

The Newton polygon of a polynomial over a valued field is the lower convex hull of the points formed from coefficient indices and valuations. Its slopes determine the valuations of roots, counted with multiplicity.

For a degree-$d$ polynomial, reflecting the usual coefficient-index axis replaces $(j,v(a_j))$ by $(d-j,v(a_j))$. The reflected lower polygon has slopes equal to root valuations, rather than their negatives. Horizontal lengths, and therefore multiplicities, are unchanged. This convention makes positive lower ramification groups breaks appear as positive slopes of a reduced ramification polygon.

For a polynomial with nonzero constant coefficient over a complete discretely valued field, plot coefficient exponents against their valuations. A lower Newton polygon segment of slope $s$ and horizontal length $r$ accounts for exactly $r$ roots of valuation $-s$, counted with multiplicity in an algebraic closure. Zero coefficients are omitted. Reflecting the horizontal axis changes the sign of every slope, so slope conventions must be fixed before reading root valuations.

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.