Algebra, Volume II

296 Pages·2003·16.82 MB·English

by  B.L. van der Waerden

B.L. van der Waerden
Algebra
Volume II
Based in part on lectures by E. Artin and E. Noether
Translated by John R. Schulenberger
Springer
B.L. van der Waerden
University of ZOrich (retired)
Present address:
Wiesliacher 5
(8053) ZOrich, Switzerland
Originally published in 1970 by Frederick Ungar Publishing Co., Inc., New York
Volume II is translated from the German Algebra II, fifth edition, Springer-Verlag Berlin, 1967.
The work was first published with the title Moderne Algebra in 1930-1931.
Mathematics Subject Classification (2000): 00A05 01A75 12-01 13-01 15-01 16-01
ISBN 0-387-40625-5 Printed on acid-free paper.
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
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PREFACE TO THE FIFfH EDITION
P. Roquette has been kind enough to provide me with a nice proof of
the residue theorem for algebraic differentials udz. The-chapter "Algebraic
Functions" has thereby been brought to a satisfactory conclusion.
In the chapter "Topological Algebra," following Bourbaki, the completion of
groups, rings, and fields has been carried out by means of filters without using
the second countability axiom.
The chapter "Linear Algebra," which is important for many applications, now
appears at the beginning of the volume, and topological algebra is treated in the
last chapter. The book now consists of three independent groups of three
chapters each:
Chapters 12-14: Linear Algebra, Algebra, Representation Theory
Chapters 15-17: Ideal Theory
Chapters 18-20: Fields with Valuations, Algebraic Functions, Topological
Algebra
This subdivision of the material is now expressed more clearly in the schematic
guide on page xv.
Zurich, March 1967 B. L. VAN DER WAERDEN
FROM THE PREFACE TO THE FOURTH EDITION
Two new chapters have been added at the beginning of the second volume:
a chapter on algebraic functions of one variable, which goes as far as the
Riemann-Roch theorem for arbitrary fields of constants, and a chapter on
topological algebra, which is mainlyconcemed with the completion of topological
groups, rings, and skew fields. I should like to thank Dr. H. R. Fischer, who
read these two chapters in manuscript form, for many useful remarks.
The chapter "General Ideal Theory" has been extended to include the impor
tant theorems ot Krull on symbolic powers of prime ideals and chains of prime
ideals. The relation of the ideal theory of integrally closed rings with valuation
theory has been brought out more clearly. A section on antisymmetric bilinear
forms has been added to the chapter "Linear Algebra."
In the chapter "Algebras," more examples are given, the theory of the radical
has been developed, following Jacobson, without a finiteness condition, and
the fundamental ideas of Emmy Noether on direct sums and intersections of
modules have been more strongly emphasized. It was possible to considerably
simplify the proofs of the principal theorems by combining the methods of
Jacobson with those of Emmy Noether.
By omitting some material I have tried to keep the size of the book within
reasonable bounds. Thus, the chapter "Elimination Theory" has been omitted.
The theorem on the existence of the resultant system for homogeneous equations,
which was formerly proved by means of elimination theory, now appears in
Section 121 as a corollary of Hilbert's Nullstellensatz.
Zurich, June 1959 B. L. VAN DER W AERDEN
GUIDE
A survey of the chapters of Volumes 1 and 2 and their logical dependence.
1
Sets
2
Groups
3
Rings
4
Vecton
S
Polynomials
9
7
Groups Infinite
Sets
6
I
Fields
8 10
Galois Infinite
Theory Fields
11
Real Fields
I I
12 18
13 IS
Linear Fields with ~
Algebras Ideal Theory
Algebra Valuations
I I I I
-I I I I I
14 17 16 19 20
Representation Integral Polynomial Algebraic Topological
Theory Elements Ideals Functions Algebra

CONTENTS
Chapter 12
LINEAR ALGEBRA 1
12.1 Modules over a Ring 1
12.2 Modules over Euclidean Rings. Elementary Divisors 3
12.3 The Fundamental Theorem of Abelian Groups 6
12.4 Representations and Representation Modules 10
12.5 Normal Forms of a Matrix in a Commutative Field 13
12.6 Elementary Divisors and Characteristic Functions 17
12.7 Quadratic and Hermitian Forms 20
12.8 Antisymmetric Bilinear Forms 28
Chapter 13
ALGEBRAS 32
13.1 Direct Sums and Intersections 33
13.2 Examples of Algebras 36
13.3 Products and Crossed Products 41
13.4 Algebras as Groups with Operators. Modules and Representations 48
13.5 The Large and Small Radicals 51
13.6 The Star Product 54
13.7 Rings with Minimal Condition 56
13.8 Two.-Sided Decompositions and Center Decomposition 60
13.9 Simple and Primitive Rings 63
13.10 The Endomorphism Ring of a Direct Sum 66
13.11 Structure Theorems for Semisimple and Simple Rings 68
13.12 The Behavior of Algebras under Extension of-the Base Field 70
Chapter 14
REPRESENTATION THEORY OF GROUPS AND ALGEBRAS 75
14.1 Statement of the Problem 75
14.2 Representation of Algebras 76
14.3 Representations of the Center 80
ix
X CONTENTS
14.4 Traces and Characters 82
14.5 Representations of Finite Groups 84
14.6 Group Characters 88
14.7 The Representations of the Symmetric Groups 93
14.8 Semigroups of Linear Transformations 97
14.9 Double Modules and Products of Algebras 99
14.10 The Splitting Fields of a Simple Algebra 105
14.11 The Brauer Group. Factor Systems 107
Chapter 15
GENERAL IDEAL THEORY OF COMMUTATIVE RINGS 115
15.1 Noetherian Rings 115
15.2 Products and Quotients of Ideals 119
15.3 Prime Ideals and Primary Ideals 122
15.4 The General Decomposition Theorem 126
15.5 The First Uniqueness Theorem 130
15.6 Isolated Components and Symbolic Powers 133
15.7 Theory of Relatively Prime Ideals 135
15.8 Single-Primed Ideals 139
15.9 Quotient Rings 141
15.10 The Intersection of all Powers of an Ideal 143
15.11 The Length of a Primary Ideal. Chains of Primary Ideals in Noetherian
Rings 145
Chapter 16
THEORY OF POLYNOMIAL IDEALS 149
16.1 Algebraic Manifolds 149
16.2 The Universal Field 151
16.3 The Zeros of a Prime Ideal 152
16.4 The Dimension 154
16.5 Hilbert's Nullstellensatz. Resultant Systems for Homogeneous Equations
156
16.6 Primary Ideals 159
16.7 Noether's Theorem 161
16.8 Reduction of Multidimensional Ideals to Zero-Dimensional Ideals
164
(7ontents xi
Chapter 17
INTEGRAL ALGEBRAIC ELEMENTS 168
17.1 Finite 9t-Modules 169
17.2 Integral Elements over a Ring 170
17.3 The Integral Elements of a Field 173
17.4 Axiomatic Foundation of Classical Ideal Theory 177
17.5 Converse and Extension of Results 180
17.6 Fractional Ideals 182
17.7 Ideal Theory of Arbitrary Integrally Closed Integral Domains 184
)
Chapter 18
FIELDS WITH VALUATIONS 191
18.1 Valuations 191
18.2 Complete Extensions 197
18.3 Valuations of the Field of Rational Numbers 201
18.4 Valuation of Algebraic Extension Fields: Complete Case 204
18.5 Valuation of Algebraic Extension Fields: General Case 210
18.6 Valuations of Algebraic Number Fields 212
18.7 Valuations of a Field d{x) of Rational Functions 217
18.8 The Approximation Theorem 221
Chapter 19
ALGEBRAIC FUNCTIONS OF ONE VARIABLE 223
19.1 Series Expansions in the Uniformizing Variable 223
19.2 Divisors and Multiples 227
19.3 The Genus g 230
19.4 Vectors and Covectors 233
19.5 Differentials. The Theorem on the Speciality Index 235
19.6 The Riemann-Roch Theorem 239
19.7 Separable Generation of Function Fields 242
19.8 Differentials and Integral~ in the Classical Case 243
19.9 Proof of the Residue Theorem 247