Algebra: Volume I

279 Pages·2003·15.43 MB·English

by  B.L. van der Waerden

B.L. van der Waerden
Algebra
Volume I
Based in part on lectures by E. Artin and E. Noether
Translated by Fred Blum and John R. Schulenberger
Springer
B.L. van der Waerden
University of ZOrich (retired)
Pres.ent address:
Wiesliacher 5
(8053) Zurich, Switzerland
Originally published in 1970 by Frederick Ungar Publishing Co., Inc., New York
Volume I is translated from the German Algebra 1, seventh edition, Springer-Verlag Berlin, 196q.
The work was first published with the title Modeme Algebra in 1930-1931.
Mathematics Subject Classification (2000): 00A05 01A75 12-01 13-01 16-01 20-01
ISBN 0-387-40624-7
First softcover printing, 2003.
© 1991 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY
10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights ..
This reprint has been authorized by Springer-Verlag (BerlinIHeidelbergINew York) for sale in
the People's Republic of China only and not for export therefrom.
987654321 SPIN 10947647
www.springer-ny.com
Springer. . Verlag New York Berlin Heidelberg
A member ofB erte/smannSpringer Science + Business Media GmbH
FOREWORD TO THE SEVENTH EDITION
When the first edition was written, it was intended as an introduction to
the newer abstract alge!>ra. Parts of classical algebra, in particular the theory
of determinants, were assumed to be known. Today, however, the book is
commonly used by students as a first introduction to algebra. It has therefore
been necessary to include a chapter on "vector spaces and tensor spaces" in which
the fundamental ideas of linear algebra, the theory of determinants in par
ticular, are discussed.
The first chapter, "Numbers and Sets," has been made shorter by treating
ordering and well ordering in a new chapter (Chapt~r 9). Zorn's lemma is
derived directly from the axiom of choice. A proof of the well ordering theorem
is obtained with the same method (following H. Kneser).
In the Galois theory certain ideas from the well-known book of Artin were
adopted. A gap in a proof in the theory of cyclic fields, which several readers
8.5.
brought to my attention, was closed in Section The existence of a normal
basis is proved in Section 8.11.
The first volume now concludes with the chapter "Real Fields." Valuation
theory is presented in the second volume.
Zurich, February 1966 B. L. VAN DER WAERDEN
FOREWORD TO THE FOURTH EDITION
The algebraist and number theorist Brandt, who recently died unexpectedly,
concluded his review of the third edition of this work in the Jahresbericht der
D. M. V. 55 as follows: "As far as the title is concerned, I w()uld welcome it if
the simpler, but more powerful title 'Algebra' were chosen for the fourth
edition. A book which offers so much of the best mathematics, as it has been
and as it will be, should not through its title give rise to the suspicion that it is
simply following a fashionable trend which yesterday was unknown and
tomorrow will probably be forgotten."
Following this suggestion, I have changed the title to "Algebra."
I am grateful for a suggestion by M. Deuring for a more appropriate definition
of the concept of a "hypercomplex system" as well as an extension of the Galois
theol7Y of cyclotomic fields~which seemed required with consideration of its
application to the theory of cyclic fields.
Many small corrections have been made on the basis of letters from various
countries. I should here like to thank all writers for their letters.
Zurich, March 1955 B. L. VAN DER W AERDEN
FROM THE PREFACE TO THE THIRD EDITION
In the second edition the treatment of valuation theory was extended. It has
meanwhile become more and more important in number theory and algebraic
geometry. I have therefore made the chapter on valuation theory clearer and
more detailed.
In response to many wishes, I have again included the section on well ordering
and transfinite induction, which was dropped in the second edition, and on this
basis again presented the Steinitz field theory in complete generality.
Following a suggestion of Zariski, the introduction of polynomials has been
made easily comprehensible. The theory of norms and traces needed improve. .
ment; Peremans had kindly pointed this out to me.
Laren (Nordholland), July 1950 B. L. VAN DER W AERDEN
GUIDE
The chapters of both volumes and their interrelation.
1
Sets
I
•
I
9
2
Infinite
Groups
Sets
3
Rinp
•
1
\
4 7
Vectors Groups
•
I 1
S
Polynomials
IS 6
Ideal Theory Fields
I
....._ ...1. . ..._ ... 1
I I I
17
8 12
13
Integral
Galois Linear
Algebraic Algebras
Theory Algebra
Quantities
-....... •... ...- -.....- ----.--..., I I
10 18 14
Infinite Fields with Representation
Fields Valuations Theory
1- •
I I I
16 11 19 20
Polynomial Real Algebraic Topological
Ideals Fields Functions Algebra

INTRODUCTION
PURPOSE OF THE BOOK
The "abstract," "formal," or "axiomatic" direction, to which the fresh impetus
in algebra is due, has led to a number of new formulations of ideas, insight into
new interrelations, and far-reaching results, especially in group theory, field
theory, valuation theory, ideal theory, and the theory of hypercomplex numbers.
The principal objective of this book is to introduce the reader into this entire
world of concepts. While, for this reason, general concepts and methods stand
in the foreground, particular results which properly belong to classical algebra
mu~t also be given appropriate co.nsideration within the framework of the
modern development.
DISTRIBUTION OF SUBJECT MATTER. DIRECTIONS FOR mE
READER
In order to develop with sufficient clarity the general viewpoints which
dominate the "abstract" approach to algebra, it was necessary to present afresh
the fundamentals of group theory and of elementary algebra.
In view of the fact that com~tent expositions on group theory, classical
algebra, and the theory of fields have been published recently,l it was possible
to present these introductory chapters briefly (but completely).
Another guiding principle was the desire to make each individual part
comprehensible in itself, insofar as this was possible. Those who wish to become
acquainted with the general theory of ideals or with the theory of hyper
complex numbers need not study Galois theory beforehand and vice versa;
those who want to consult the book about elimination or linear algebra need
not be deterred by complicated ideal-theoretical terms.
For this reason the subject matter has been distributed in such fashion that
the first three chapters contain a most concise exposition of what is prerequisite
to'all subsequent chapters: The fundamentals of the theories of 1. sets, 2. groups,
IFor group theory the reader is referred to: Speiser, A.: Die Theorie der Gruppen von endlicher
Ordnung, 2nd edition, Berlin. Julius Springer 1927
For the theory of fields: Hasse, H.: Hohere Algebra I, II and Aufgabensammlung zur Hoheren
Algebra. Sammlung Goschen 1926-27.
For classical algebra: Perron, 0.: Algebra I, II. 1927.
For linear algebra: Dickson, L. E.: Modern Algebraic Theories, Chicago 1926.
X INTRODUcnON
3. rings, ideals, and fields. The remaining chapters of the first volume are in the
main devoted to the theory of commutative fields and are based primarily on
Steinitz' fundamental treatise in erellea Journal, Vol. 137 (1910). The theory of
modules, rings, and ideals with applications to algebraic functions, elementary
divisors, hypercomplex numbers, and group representations will be treated in
the second volume in several, mostly independent chapters.
The theory of abelian integrals and the theory of continuous groups had to be
omitted, since an appropriate treatment of both involves transcendental concepts
and methods. Because ofi ts extent, the theory of invariants could not be included,
either.
For further information we refer the reader to the table of contents, and
especially to the foregoing schematic diagram which illustrates exactly how many
of the preceding chapters are requisite to each of the chapters.
The interspersed exercises may serve as a test whether the subject has become
clear to the reader. Some of them contain examples and supplements, which are
sometimes referred to in later chapters. No special devices are necessary for
their solutions unless indicated in square brackets.
SOURCES
This book has in part grown out of the following courses:
Lectures given by E. Artin on Algebra (Hamburg, Summer session 1926).
A seminar on Theory of Ideals, conducted by E. Artin, W. Blaschke, O.
Schreier, and the author (Hamburg, Winter 1926-27).
Lectures by E. Noether on Group Theory and Hypercomplex Numbers
(Gattingen, Winter 1924-25 and Winter 1927-28)1.
New proofs or new arrangements of proofs in this book are in most cases due
to the lectures and seminars mentioned, regardless of whether the source is
expressly quoted.
IAn elaborate treatment of the latter course by E. NoEtHER was published in MatA. Zeitschri/t
Vol. 30 (1929), pp. 641-692.
CONTENTS
•
Chapter 1
NUMBERS AND SETS 1
1.1 Sets 1
1.2 Mappings. Cardinality 2
1.3 The Number Sequence 3
1.4 Finite and Countable (Denumerable) Sets 7
1.5 Partitions 10
Chapter 2
GROUPS 12
2.1 The Concept of a Group 12
2.2 SubgrQups 19
2.3 Complexes. Cosets 23
2.4 Isomorphisms and Automorphisms 25
2.S Homomorphisms, Normal Subgroups, and Factor Groups 28
Chapter 3
RINGS AND FIELDS 32
3.1 Rings 32
3.2 Homomorphism and Isomorphism 39
3.3 The Concept of a Field of Quotients 40
3.4 Polynomial Rings 43
3.5 Ideals. Residue Class Rings 47
3.6 Divisibility. Prime Ideals 51
3.7 Euclidean Rings and Principal Ideal Rings 53
3.8 Factorization 57