In this chapter, we recall the traditional theory of Newton polygons for polynomials over a nonarchimedean field. In the process, we introduce a general framework which will allow us to consider Newton polygons in a wider range of circumstances; it is based on a version of Hensel's lemma that applies in not necessarily commutative rings. As a first application, we fill in a few missing proofs from Chapter 1.

Introduction to Newton polygons

We start with the possibly familiar notion of the Newton polygon associated with a polynomial over a nonarchimedean ring.

Definition 2.1.1. Let R be a ring equipped with a nonarchimedean submultiplicative (semi)norm | · |. For ρ > 0 and P = Σi Pi Ti ∈ R[T], define the width of P under the ρ-Gauss norm | · |ρ as the difference between the maximum and minimum values of i for which maxi{|Pi|ρi} is achieved.

Proposition 2.1.2.Let R be a ring equipped with a nonarchimedean multiplicative seminorm | · |. For ρ > 0 and P, Q ∈ R[T] the following results hold.

1. (a) We have |PQ|ρ = |P|ρ|Q|ρ. That is, | · |ρis multiplicative.

2. (b) The width of PQ under | · |ρequals the sum of the widths of P and Q under | · |ρ.

Proof. For * ∈ {P, Q}, let j*, k* be the minimum and maximum values of i for which maxi {|*i| ρi} is achieved.