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

Abstract

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer.

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.…”

Reduced normal form of local integral bases

“…This is the case, for instance, when L/K is separable, or K is complete, or A is a finitely generated algebra over a field [15,Ch.I,§4].…”

“…, ν n are known. Such a basis is provided, for instance, by the method of the quotients [6], or the multipliers method [1], both based on the Montes algorithm [5,4].…”

Okutsu-Montes representations of prime ideals of one-dimensional integral closures

“…Let us sketch the application of Montes algorithm [HN08], [GMN08] (HN stands for "higher Newton"). The design of the algorithm will be described in more detail in Section 2.…”