BASIC ALGEBRA II Second Edition Nathan Jacobson Yale University Dover Publications, Inc. Mineola, New York To Mike and Polly Copyright Copyright © 1980, 1989 by Nathan Jacobson All rights reserved. Bibliographical Note This Dover edition, first published in 2009, is an unabridged republication of the 1989 second edition of the work originally published by W. H. Freeman and Company, San Francisco, in 1980. Library of Congress Cataloging-in-Publication Data Jacobson, Nathan, 1910–1999. Basic algebra / Nathan Jacobson. — Dover ed. p. cm. Originally published: 2nd ed. San Francisco : W.H. Freeman, 1985–1989. ISBN-13: 978-0486-47189-1 (v. 1) ISBN-10: 0-486-47189-6 (v. 1) ISBN 13: 978-0-486-47187-7 (v. 2) ISBN-10 0-486-47187-X (v. 2) 1. Algebra I. Title. QA154.2.J32 2009 512.9—dc22 2009006506 Manufactured in the United States by Courier Corporation 47187X01 www.doverpublications.com Contents Contents of Basic Algebra I Preface Preface to the First Edition INTRODUCTION 0.1 Zorn’s lemma 0.2 Arithmetic of cardinal numbers 0.3 Ordinal and cardinal numbers 0.4 Sets and classes References 1 CATEGORIES 1.1 Definition and examples of categories 1.2 Some basic categorical concepts 1.3 Functors and natural transformations 1.4 Equivalence of categories 1.5 Products and coproducts 1.6 The horn functors. Representable functors 1.7 Universals 1.8 Adjoints References 2 UNIVERSAL ALGEBRA 2.1 Ω-algebras 2.2 Subalgebras and products 2.3 Homomorphisms and congruences 2.4 The lattice of congruences. Subdirect products 2.5 Direct and inverse limits 2.6 Ultraproducts 2.7 Free Ω-algebras 2.8 Varieties 2.9 Free products of groups 2.10 Internal characterization of varieties References 3 MODULES 3.1 The categories R-mod and mod-R 3.2 Artinian and Noetherian modules 3.3 Schreier refinement theorem. Jordan-Hölder theorem 3.4 The Krull-Schmidt theorem 3.5 Completely reducible modules 3.6 Abstract dependence relations. Invariance of dimensionality 3.7 Tensor products of modules 3.8 Bimodules 3.9 Algebras and coalgebras 3.10 Projective modules 3.11 Injective modules. Injective hull 3.12 Morita contexts 3.13 The Wedderburn-Artin theorem for simple rings 3.14 Generators and progenerators 3.15 Equivalence of categories of modules References 4 BASIC STRUCTURE THEORY OF RINGS 4.1 Primitivity and semi-primitivity 4.2 The radical of a ring 4.3 Density theorems 4.4 Artinian rings 4.5 Structure theory of algebras 4.6 Finite dimensional central simple algebras 4.7 The Brauer group 4.8 Clifford algebras References 5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS 5.1 Representations and matrix representations of groups 5.2 Complete reducibility 5.3 Application of the representation theory of algebras 5.4 Irreducible representations of S n 5.5 Characters. Orthogonality relations 5.6 Direct products of groups. Characters of abelian groups 5.7 Some arithmetical considerations 5.8 Burnside’s paqb theorem 5.9 Induced modules 5.10 Properties of induction. Frobenius reciprocity theorem 5.11 Further results on induced modules 5.12 Brauer’s theorem on induced characters 5.13 Brauer’s theorem on splitting fields 5.14 The Schur index 5.15 Frobenius groups References 6 ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS 6.1 Additive and abelian categories 6.2 Complexes and homology 6.3 Long exact homology sequence 6.4 Homotopy 6.5 Resolutions 6.6 Derived functors 6.7 Ext 6.8 Tor 6.9 Cohomology of groups 6.10 Extensions of groups 6.11 Cohomology of algebras 6.12 Homological dimension 6.13 Koszul’s complex and Hilbert’s syzygy theorem References 7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS 7.1 Prime ideals. Nil radical 7.2 Localization of rings 7.3 Localization of modules 7.4 Localization at the complement of a prime ideal. Local-global relations 7.5 Prime spectrum of a commutative ring 7.6 Integral dependence 7.7 Integrally closed domains 7.8 Rank of projective modules 7.9 Projective class group 7.10 Noetherian rings 7.11 Commutative artinian rings 7.12 Affine algebraic varieties. The Hilbert Nullstellensatz 7.13 Primary decompositions 7.14 Artin-Rees lemma. Krull intersection theorem 7.15 Hilbert’s polynomial for a graded module 7.16 The characteristic polynomial of a noetherian local ring 7.17 Krull dimension 7.18 I-adic topologies and completions References 8 FIELD THEORY 8.1 Algebraic closure of a field 8.2 The Jacobson-Bourbaki correspondence 8.3 Finite Galois theory 8.4 Crossed products and the Brauer group 8.5 Cyclic algebras 8.6 Infinite Galois theory 8.7 Separability and normality 8.8 Separable splitting fields 8.9 Kummer extensions 8.10 Rings of Witt vectors 8.11 Abelian p-extension 8.12 Transcendency bases 8.13 Transcendency bases for domains. Affine algebras 8.14 Luroth’s theorem 8.15 Separability for arbitrary extension fields 8.16 Derivations 8.17 Galois theory for purely inseparable extensions of exponent one 8.18 Tensor products of fields 8.19 Free composites of fields References 9 VALUATION THEORY 9.1 Absolute values 9.2 The approximation theorem 9.3 Absolute values on and F(x) 9.4 Completion of a field 9.5 Finite dimensional extensions of complete fields. The archimedean case 9.6 Valuations 9.7 Valuation rings and places 9.8 Extension of homomorphisms and valuations 9.9 Determination of the absolute values of a finite dimensional extension field 9.10 Ramification index and residue degree. Discrete valuations 9.11 Hensel’s lemma 9.12 Local fields 9.13 Totally disconnected locally compact division rings 9.14 The Brauer group of a local field 9.15 Quadratic forms over local fields References 10 DEDEKIND DOMAINS 10.1 Fractional ideals. Dedekind domains 10.2 Characterizations of Dedekind domains 10.3 Integral extensions of Dedekind domains 10.4 Connections with valuation theory 10.5 Ramified primes and the discriminant 10.6 Finitely generated modules over a Dedekind domain References 11 FORMALLY REAL FIELDS 11.1 Formally real fields 11.2 Real closures 11.3 Totally positive elements 11.4 Hilbert’s seventeenth problem 11.5 Pfister theory of quadratic forms 11.6 Sums of squares in R(x ,…,x ), R a real closed field 1 n 11.7 Artin-Schreier characterization of real closed fields References INDEX