Newton polygons of higher order in algebraic number theory

Abstract

We develop a theory of arithmetic Newton polygons of higher\ud order, that provides the factorization of a separable polynomial over a p-adic\ud eld, together with relevant arithmetic information about the elds generated\ud by the irreducible factors. This carries out a program suggested by . Ore.\ud As an application, we obtain fast algorithms to compute discriminants, prime\ud ideal decomposition and integral bases of number elds.Postprint (author’s final draft

Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

“…The theory of higher order Newton polygons developed in [12], and revised in [7], has revealed itself as a powerful tool for the analysis of the decomposition of a prime p in a number field. Newton polygons of higher order are a p-adic tool, and their computation involves no extension of the ground field but only extensions of the residue field; thus, they constitute an excellent device for a computational treatment of the problem.…”

“…Newton polygons of higher order are a p-adic tool, and their computation involves no extension of the ground field but only extensions of the residue field; thus, they constitute an excellent device for a computational treatment of the problem. In this paper we explain how the theoretical results of [7] apply to yield an algorithm, due to the second author [12,Ch.3], to factor a prime number p in a number field K, in terms of a generating polynomial f (x). The algorithm computes the p-valuation of the index of f (x) as well; in particular, it determines the discriminant of the number field, if one is able to factorize the discriminant disc(f ) of the defining equation.…”

“…In section 2 we present the main technical ingredients of the paper: types and their representatives, and we review the basic algorithm that is obtained by a direct application of the ideas of [7]. In section 3 we characterize the optimal choices of representatives (Theorem 3.1).…”

Discriminants of Chebyshev radical extensions

“…Our proof relies on the work of Guàrdia, Montes, and Nart [9,10,11]. In particular, the computation of the index comes from an analysis of specialized Newton polygons, which we describe in Section 4.…”

“…We compute ind(Φ) using a relatively recent algorithm derived by Guàrdia, Montez, and Nart [9,10,11]. Their method employs a more refined variation of the Newton polygon, called the φ-Newton polygon, which captures arithmetic data attached to each irreducible factor φ of Φ.…”