Basic Algebra II (Nathan Jacobson)

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Transcript of Basic Algebra II (Nathan Jacobson)

Basic Algebra IISecond EditionNATHAN JACOBSONYALE U N I V E R S I T Y

W . H. F R E E M A N A N D N e w York

B

COMPANY

To Mike and Polly

Library of Congress Cataloging-in-Publication Data (Revised for vol. 2) Jacobson, Nathan, 1910Basic algebra. Includes bibliographical references and indexes. 1. Algebra. I. Tide. QA154.2.J32 1985 512.9 84-25836 ISBN 0-7167-1480-9 (v. 1) ISBN 0-7167-1933-9 (v. 2) Copyright 1989 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 VB 7 6 5 4 3 2 1 0 8 9

Contents

Contents of Basic AlgebraPreface xiii

I

x

Preface to the First Edition

xv

INTRODUCTION

1

0.1 0.2 0.3 0.4

Zorn's lemma 2 Arithmetic of cardinal numbers 3 Ordinal and cardinal numbers 4 Sets and classes 6 References 7

1

CATEGORIES

8

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Definition and examples of categories 9 Some basic categorical concepts 15 Functors and natural transformations 18 Equivalence of categories 26 Products and coproducts 32 The horn functors. Representable functors 37 Universals 40 Adjoints 45 References 51

2

UNIVERSAL ALGEBRA

52

2.1 2.2 2.3 2.4

ft-algebras 53 Subalgebras and products 58 Homomorphisms and congruences 60 The lattice of congruences. Subdirect products

66

vi

Conten

2.5 2.6 2.7 2.8 2.9 2.10

Direct and inverse limits 70 Ultraproducts 75 Free Q-algebras 78 Varieties 81 Free products of groups 87 Internal characterization of varieties 91 References 93

3

MODULES

94

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15

The categories R-mod and mod-R 95 Artinian and Noetherian modules 100 Schreier refinement theorem. Jordan-Holder theorem 104 The Krull-Schmidt theorem 110 Completely reducible modules 117 Abstract dependence relations. Invariance of dimensionality Tensor products of modules 125 Bimodules 133 Algebras and coalgebras 137 Projective modules 148 Injective modules. Injective hull 156 Morita contexts 164 The Wedderburn-Artin theorem for simple rings 171 Generators and progenerators 173 Equivalence of categories of modules 177 References 183

122

4

B A S I C S T R U C T U R E T H E O R Y OF RINGS

184

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Primitivity and semi-primitivity 185 The radical of a ring 192 Density theorems 197 Artinian rings 202 Structure theory of algebras 210 Finite dimensional central simple algebras 215 The Brauer group 226 Clifford algebras 228 References 245

5

CLASSICAL REPRESENTATION THEORY OF FINITE G R O U P S 246

5.1 5.2 5.3 5.4

Representations and matrix representations of groups 247 Complete reducibility 251 Application of the representation theory of algebras 257 Irreducible representations of S 265n

Contents

vii

5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

Characters. Orthogonality relations 269 Direct products of groups. Characters of abelian groups 279 Some arithmetical considerations 282 Burnside's p q theorem 284 Induced modules 286 Properties of induction. Frobenius reciprocity theorem 292 Further results on induced modules 299 Brauer's theorem on induced characters 305 Brauer's theorem on splitting fields 313 The Schur index 314 Frobenius groups 317 References 325a b

ELEMENTS OF H O M O L O G I C A L A L G E B R A WITH APPLICATIONS 326

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13

Additive and abelian categories 327 Complexes and homology 331 Long exact homology sequence 334 Homotopy 337 Resolutions 339 Derived functors 342 Ext 346 Tor 353 Cohomology of groups 355 Extensions of groups 363 Cohomology of algebras 370 Homological dimension 375 Koszul's complex and Hilbert's syzygy theorem References 387

378

7

C O M M U T A T I V E IDEAL THEORY: GENERAL THEORY A N D NOETHERIAN RINGS 388

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

Prime ideals. Nil radical 389 Localization of rings 393 Localization of modules 397 Localization at the complement of a prime ideal. Local-global relations 400 Prime spectrum of a commutative ring 403 Integral dependence 408 Integrally closed domains 412 Rank of projective modules 414 Projective class group 419 Noetherian rings 420 Commutative artinian rings 425

viii

Contents

7.12 7.13 7.14 7.15 7.16 7.17 7.18

Affine algebraic varieties. The Hilbert Nullstellensatz 427 Primary decompositions 433 Artin-Rees lemma. Krull intersection theorem 440 Hilbert's polynomial for a graded module 443 The characteristic polynomial of a noetherian local ring 448 Krull dimension 450 J-adic topologies and completions 455 References 462463

8

FIELD T H E O R Y

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19

Algebraic closure of a field 464 The Jacobson-Bourbaki correspondence 468 Finite Galois theory 471 Crossed products and the Brauer group 475 Cyclic algebras 484 Infinite Galois theory 486 Separability and normality 489 Separable splitting fields 495 Kummer extensions 498 Rings of Witt vectors 501 Abelian p-extension 509 Transcendency bases 514 Transcendency bases for domains. Affine algebras 517 Luroth's theorem 520 Separability for arbitrary extension fields 525 Derivations 530 Galois theory for purely inseparable extensions of exponent one 541 Tensor products of fields 544 Free composites of fields 550 References 556557

9

VALUATION THEORY

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

Absolute values 558 The approximation theorem 562 Absolute values on Q and F(x) 564 Completion of a field 566 Finite dimensional extensions of complete fields. The archimedean case 569 Valuations 573 Valuation rings and places 577 Extension of homomorphisms and valuations 580 Determination of the absolute values of a finite dimensional extension field 585 Ramification index and residue degree. Discrete valuations 588 Hensel's lemma 592 Local fields 595

Contents

ixTotally disconnected locally compact division rings 599 The Brauer group of a local field 608 Quadratic forms over local fields 611 References 618619

9.13 9.14 9.15

10

DEDEKIND D O M A I N S

10.1 10.2 10.3 10.4 10.5 10.6

Fractional ideals. Dedekind domains 620 Characterizations of Dedekind domains 625 Integral extensions of Dedekind domains 631 Connections with valuation theory 634 Ramified primes and the discriminant 639 Finitely generated modules over a Dedekind domain References 649650

643

11

F O R M A L L Y REAL FIELDS

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Formally real fields 651 Real closures 655 Totally positive elements 657 Hilbert's seventeenth problem 660 Pfister theory of quadratic forms 663 Sums of squares in R(x ..., x ), R a real closed field 669 Artin-Schreier characterization of real closed fields 674 References 677u n

INDEX

679

Contents of Basic A l g e b r a I

I N T R O D U C T I O N : C O N C E P T S F R O M SET T H E O R Y . THE INTEGERS 1

0.1 0.2 0.3 0.4 0.5 0.6 0.71

The power set of a set 3 The Cartesian product set. Maps 4 Equivalence relations. Factoring a map through an equivalence relation The natural numbers 15 The number system Z of integers 19 Some basic arithmetic facts about Z 22 A word on cardinal numbers 2426

10

MONOIDS A N D GROUPS

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.132

Monoids of transformations and abstract monoids 28 Groups of transformations and abstract groups 31 Isomorphism. Cayley's theorem 36 Generalized associativity. Commutativity 39 Submonoids and subgroups generated by a subset. Cyclic groups 42 Cycle decomposition of permutations 48 Orbits. Cosets of a subgroup 51 Congruences. Quotient monoids and groups 54 Homomorphisms 58 Subgroups of a homomorphic image. Two basic isomorphism theorems 64 Free objects. Generators and relations 67 Groups acting on sets 71 Sylow's theorems 7985

RINGS

2.1 2.2 2.3 2.4 2.5

Definition and elementary properties Types of rings 90 Matrix rings 92 Quaternions 98 Ideals' quotient rings 101

86

Contents

xiIdeals and quotient rings for Z 103 Homomorphisms of rings. Basic theorems 106 Anti-isomorphisms 111 Field of fractions of a commutative domain 115 Polynomial rings 119 Some properties of polynomial rings and applications 127 Polynomial functions 134 Symmetric polynomials 138 Factorial monoids and rings 140 Principal ideal domains and Euclidean domains 147 Polynomial extensions of factorial domains 151 "Rings" (rings without unit) 155

2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17

3

M O D U L E S OVER A P R I N C I P A L I D E A L D O M A I N

157

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Ring of endomorphisms of an abelian group 158 Left and right modules 163 Fundamental concepts and results 166 Free modules and matrices 170 Direct sums of modules 175 Finitely generated modules over a p.i.d. Preliminary results 179 Equivalence of matrices with entries in a p.i.d. 181 Structure theorem for finitely generated modules over a p.i.d. 187 Torsion modules' primary components' invariance theorem 189 Applications to abelian groups and to linear transformations 194 The ring of endomorphisms of a finitely generated module over a p.i.d. 204