Computing local Artin maps, and solvability of norm equations
type: Item
instance of: scholarly article
title: Computing local Artin maps, and solvability of norm equations
zbMATH Open document ID: 0999.11077
author: Vincenzo Acciaro
author: Jürgen Klüners
published in: Journal of Symbolic Computation
publication date: 2001-02-28T00:00:00Z
review text: Based on the efficient computation of the automorphism group of abelian extensions of number fields by the authors [Math. Comput. 68, 1179-1186 (1999; Zbl 0937.11062)] and \textit{J. Klüners} [Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper. Berlin: TU Berlin (1997; Zbl 0912.11059)] they apply the results to the task of computing local and global Artin symbols. As an application they use Hasse's norm theorem to check for solvability of norm equations in cyclic extensions. In the previous works the authors computed the automorphism group of abelian extensions by explicitly constructing Frobenius automorphisms corresponding to certain unramified primes. Once the full group of automorphisms is known, it is straightforward to identify the Frobenius automorphism for any unramified prime ideal. Using some class field theory, the authors show how this knowledge of the global Artin correspondence enables them to compute arbitrary (ramified) local Artin maps. In order to apply this to the solvability of norm equations in cyclic extensions, they utilise Hasse's norm theorem stating that solvability of the norm equation is equivalent to the local solvability which is equivalent to the vanishing of the local Artin symbols. As a reduction step they show that only a finite number of local symbols have to be considered. They demonstrate the efficiency in several examples using Kash [KANT V4. J. Symb. Comput. 24, 267-283 (1997; Zbl 0886.11070)]. Since the previously implemented norm equation solver is particularly slow in the case on unsolvable equations, this greatly enhances the performance.
reviewed by: Claus Fieker
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11Y50
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1532921
zbMATH Keywords: automorphism group
zbMATH Keywords: Artin symbol
zbMATH Keywords: Frobenius automorphism
zbMATH Keywords: norm equations
zbMATH Keywords: Hasse norm theorem
describes a project that uses: KANT/KASH
MaRDI profile type: Publication
OpenAlex ID: W2091344887
cites work: Solvability of norm equations over cyclic number fields of prime degree
cites work: Computing automorphisms of abelian number fields
cites work: Q3139838
cites work: Hermite and Smith normal form algorithms over Dedekind domains
cites work: Efficient rational number reconstruction
cites work: KANT V4
cites work: Algorithms in Algebraic Number Theory
cites work: Q4298518
cites work: Q3714149
cites work: Q4692759
DOI: 10.1006/JSCO.2000.0361
description: scientific article; zbMATH DE number 1532921