Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields
type: Item
instance of: scholarly article
title: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields
zbMATH Open document ID: 1266.11131
author: Jesús Montes
author: Jordi Guárdia
author: Enric Nart
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2012-08-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/0807.4065
full work available at URL: https://eudml.org/doc/219682
review text: Let \(K\) be a number field, let \(\alpha\) be an element of the ring of integers \({\mathbb Z}_K\) of \(K\) such that \(K={\mathbb Q}(\alpha)\), and let \(f(X)\in{\mathbb Z}[X]\) be the minimum polynomial of \(\alpha\) over \(\mathbb Q\). A basic problem in computational number theory is determining the factorization of \(p{\mathbb Z}_K\) into prime ideals. This paper describes an algorithm for computing this factorization in terms of \(f(X)\). The algorithm is based on the theory of higher order Newton polygons which was described in [Trans. Am. Math. Soc. 364, No. 1, 361--416 (2012; Zbl 1252.11091)]; in this paper the procedures from the cited article are refined to improve the running time. The algorithm produces generators for each of the prime ideals in \({\mathbb Z}_K\) lying over \(p{\mathbb Z}\), and expresses \(p{\mathbb Z}_K\) as a product of powers of these ideals. It also determines the \(p\)-adic valuation of the index of \({\mathbb Z}[\alpha]\) in \({\mathbb Z}_K\), and thus gives a method for computing the discriminant of \(K\). The procedures that are described here have been implemented by the authors and applied successfully in some examples involving number fields of fairly high degree.
reviewed by: Kevin Keating
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 6075052
zbMATH Keywords: Newton polygons
zbMATH Keywords: discriminants
zbMATH Keywords: ideal decomposition
MaRDI profile type: Publication
OpenAlex ID: W2963254673
arXiv ID: 0807.4065
cites work: Approximating rings of integers in number fields
cites work: Q4488163
cites work: The construction of maximal orders over a Dedekind domain
cites work: Implementing the Round Four maximal order algorithm
cites work: A fast algorithm for polynomial factorization over \(\mathbb Q_p\)
cites work: On the Complexity of the Montes Ideal Factorization Algorithm
cites work: Newton polygons of higher order in algebraic number theory
cites work: Higher Newton polygons and integral bases
cites work: Okutsu invariants and Newton polygons
cites work: A new computational approach to ideal theory in number fields
cites work: Single-factor lifting and factorization of polynomials over local fields
cites work: Construction of integral basis. II
cites work: Factoring Polynomials over Local Fields II
cites work: Q5534341
cites work: Q5661986
DOI: 10.5802/JTNB.782
Recommended article: Newton polygons of higher order in algebraic number theory
Recommended article: Polynomial factorization algorithms over number fields
Recommended article: Factoring Multivariate Polynomials over Algebraic Number Fields
Recommended article: Intrinsic factorization of ideals in Dedekind domains
Recommended article: Q4238960
Recommended article: Q3735804
Recommended article: Factoring polynomials over local fields.
Recommended article: Factoring Polynomials over Algebraic Number Fields
Recommended article: Q3801671
Recommended article: Q3327715
description: scientific article; zbMATH DE number 6075052