Cubic fields with geometry
type: Item
instance of: scholarly article
title: Cubic fields with geometry
zbMATH Open document ID: 1451.11001
DOI: 10.1007/978-3-030-01404-9
author: Samuel A. Hambleton
author: Hugh C. Williams
published in: CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
publication date: 2018-10-29T00:00:00Z
review text: This book is a first modern presentation of the theory of cubic fields, stressing its algorithmic and computational features. The only previous monograph on this subject has been published sixty years ago by [\textit{B. N. Delone} and \textit{D. K. Faddeev}, Tr. Mat. Inst. Steklova 11, 340 p. (1940; Zbl 0061.09001)]. The first chapter provides an introduction to the principal notions of the theory of cubic fields (discriminant, integral basis, orders, units, \dots) and presents explicit formulas for integral bases due to \textit{G. Woronoi} [On the algebraic integers which depend on a root of an equation of third degree. St. Petersburg (1894; JFM 25.0302.02); Collected works. In 3 Vols. Vols. I, II (Russian). Kiev: Verlag der Akademie der Wissenschaften der Ukrainischen SSR (1952; Zbl 0049.02804)] and \textit{K. Belabas} [Math. Comput. 66, No. 219, 1213--1237 (1997; Zbl 0882.11070)]. In chapter 2 one finds the theory of ideals in cubic fields, containing a description of factorization of rational primes into prime ideals based on the results of \textit{P. Llorente} and \textit{E. Nart} [Proc. Am. Math. Soc. 87, 579--585 (1983; Zbl 0514.12003)]. This leads to a formula for the field discriminant. The chapter ends with information about the class-number formula and bounds for \(\kappa\), the residue of the Dedekind zeta-function at \(s=1\). Chapter 3 is devoted to binary cubic forms, including their reduction and composition, based on modern approach to the ideas of \textit{G. Eisenstein} [Mathematische Werke. I, II. New York: Chelsea (1975; Zbl 0339.01018)]. It is shown how the theory of equivalence classes of such forms of a given discriminant under the action of \(\mathrm{SL}(2,\mathbb Z)\) influences various algebraic properties of the corresponding cubic fields, and this is applied to tabulation of such fields. An important technical role play here the notions of Hessian quadratic forms \({\mathcal Q}(x,y)\) and Jacobian cubic forms \({\mathcal F}(x,y)\) associat…
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D59
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11Y65
Mathematics Subject Classification ID: 14Q10
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 6967934
zbMATH Keywords: cubic fields
zbMATH Keywords: cubic Pell equations
zbMATH Keywords: cubic forms
zbMATH Keywords: continued fractions of cubic irrationalities
zbMATH Keywords: cubic units
reviewed by: Władysław Narkiewicz
describes a project that uses: Mathematica
describes a project that uses: LMFDB
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-030-01404-9
OpenAlex ID: W2900029085
Recommended article: A fast algorithm to compute cubic fields
Recommended article: Q4375593
Recommended article: Q4377089
Recommended article: Q4213390
Recommended article: Computations in Cubic Function Fields of Characteristic Three
Recommended article: On quadratic fields with large 3-rank
Recommended article: Q3476909
Recommended article: Computation of the Class Number and Class Group of a Complex Cubic Field
Recommended article: A Table of Fundamental Pairs of Units in Totally Real Cubic Fields
Recommended article: Q5838513
description: scientific article; zbMATH DE number 6967934