Number theory. An introduction to algebra
type: Item
instance of: scholarly article
title: Number theory. An introduction to algebra
zbMATH Open document ID: 0865.11001
author: Armin Leutbecher
published in: Springer-Lehrbuch
publication date: 1996-10-31T00:00:00Z
review text: The book under review gives an introduction to algebra by means of number theory. The author mentions the XI. supplement of Dedekind to the book ``Vorlesungen über Zahlentheorie'' of Dirichlet as an early model for such a procedure. In fact, the XI. supplement is one of the most important sources of contemporary algebra. The book begins with elementary number theory including the quadratic reciprocity law for the Jacobi symbol, quadratic number fields, continued fractions, \(p\)-adic numbers and Hilbert's norm symbol (Chapters 1-10). In Chapter 11 a short treatment of group theory, containing notions and results necessary for algebraic number theory, which, as arithmetics of algebraic number fields, is developed in Chapters 12 and 13. The author comes back to algebra in Chapters 14 and 15, where one finds a treatment of Galois theory following the method of E. Artin. The Galois theory is then used in Chapter 16 to develop the theory of different and discriminant for number fields and in Chapter 17, the theory of abelian extensions of \(\mathbb{Q}\) including the theorem of Kronecker and Weber. Minkowski's geometry of numbers follows and only now the unit theorem of Dirichlet. Chapter 18 contains, besides these topics, a treatment of Euclidean fields following H. W. Lenstra. Chapter 19 contains something of analytic number theory and the book ends with a consideration of the valuations of number fields. The book is well written and contains, besides standard material from algebra, an introduction to algebraic number theory on the basis of the ideas of Dedekind. Hensel's method of \(p\)-adic number fields is not really used. Nevertheless, the book is full of beautiful results and can be recommended for mathematicians who have some knowledge in algebra and want to become acquainted with Galois theory and classical algebraic number theory.
Mathematics Subject Classification ID: 11-01
Mathematics Subject Classification ID: 12-01
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11A15
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 940599
zbMATH Keywords: \(p\)-adic numbers
zbMATH Keywords: introduction to algebra
zbMATH Keywords: elementary number theory
zbMATH Keywords: quadratic reciprocity law for the Jacobi symbol
zbMATH Keywords: quadratic number fields
zbMATH Keywords: continued fractions
zbMATH Keywords: Hilbert's norm symbol
zbMATH Keywords: group theory
zbMATH Keywords: algebraic number fields
zbMATH Keywords: Galois theory
zbMATH Keywords: different
zbMATH Keywords: discriminant
zbMATH Keywords: abelian extensions
zbMATH Keywords: geometry of numbers
zbMATH Keywords: unit theorem of Dirichlet
zbMATH Keywords: Euclidean fields
zbMATH Keywords: analytic number theory
zbMATH Keywords: valuations
reviewed by: Helmut Koch
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 940599