Okutsu invariants and Newton polygons

摘要

Let K be a local field with perfect residue class field, its ring of integers, m the maximal ideal of O, and v : K~*→ Q the canon-ical extension of the discrete valuation of K to an algebraic closure of K. Let F(x) ∈ O[x] be a monic irreducible polynomial, θ ∈K a root of F(x), and L = K(θ). Kosaku Okutsu [Oku] attached to F(x) a family of monic irreducible separable polynomials, F_1(x), , F_r(x) ∈O[x], called the itive divisor polynomials of F(x). Take Fo(x) = 1. For each 1 < i ≤ r, deg F_i is minimal among all monic irreducible polynomials g(x) ∈ [x] satisfying υ(g(θ))/deg g > υ(F_(i-1)(θ))/deg F_(i-1), and υ(F_i(θ)) is maximal among all polynomials having this minimal degree. Let us call the chain [F_1, , F_r] an Okutsu frame of F(x), and let K_1, , K_r be the respective extensions of K determined by these polynomials. The polynomials F_1, , F_r are not uniquely determined, but many of their invariants, like the residual degrees f (K_i / K) and the ramification indices e(K_i/ K), depend only on F(x), and they are linked to some arithmetical invariants of the extension L / K and its subextensions (Corollaries 2.8 and 2.9).