Basic algebra : groups, rings and fields

478 Pages·2005·45.96 MB·English

by  Cohn

Basic Algebra

P.M. Cohn
Basci Algebra
Groups, Rings and Fields
m Springer
P.M. Cohn, MA, PhD, FRS
Department of Mathematics, University College London,
Gower Street, London WC1E 6BT, UK
British Library Cataloguing in Publication Data
Cohn, P. M. (Paul Moritz)
Basic algebra: groups, rings and fields
I. Algebra 2. Rings (Algebra) 3. Algebraic fields
I. Title
512
ISBN 978-1-4471-1060-6
Library of Congress Cataloging-in-Publication Data
Cohn, P.M. (Paul Moritz)
Basic algebra: groups, rings, and fields/P.M. Cohn,
p. cm.
Includes bibliographical references and indexes.
ISBN 978-1-4471-1060-6 ISBN 978-0-85729-428-9 (eBook)
DOI 10.1007/978-0-85729-428-9
1. Algebra. I. Title.
QA154.3.C64 2002
512-dc21 2002070686
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
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or in the case of reprographic reproduction in accordance with the terms of licences issued by the
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publishers.
springeronline.com
© Professor P.M. Cohn 2003
Originally published by Springer-Verlag London Berlin Heidelberg in 2003
Softcover reprint of the hardcover 1st edition 2003
2nd printing 2005
The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a
specific statement, that such names are exempt from the relevant laws and regulations and therefore free
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The publisher makes no representation, express or implied, with regard to the accuracy of the information
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Typesetting by BC Typesetting, Bristol BS31 1NZ
12/3830-54321 Printed on acid-free paper SPIN 11329916
Contents
Preface ...................................................................................................................... ix
Conventions on Terminology ................................................................................ xi
1. Sets
1.1 Finite, Countable and Uncountable Sets ........................................... . 1
1.2 Zorn's Lemma and Well-ordered Sets ............................................... . 8
1.3 Graphs ................................................................................................... . 15
2. Groups
2.1 Definition and Basic Properties........................................................... 25
2.2 Permutation Groups ............................................................................. 32
2.3 The Isomorphism Theorems......................... ....................................... 34
2.4 Soluble and Nilpotent Groups............................................................. 37
2.5 Commutators......................................................................................... 42
2.6 The Frattini Subgroup and the Fitting Subgroup ............................. 46
3. Lattices and Categories
3.1 Definitions; Modular and Distributive Lattices ................................. 51
3.2 Chain Conditions .................................................................................. 60
3.3 Categories............................................................................................... 65
3.4 Boolean Algebras................................................................................... 70
4. Rings and Modules
4.1 The Definitions Recalled .......................... ............................................ 79
4.2 The Category of Modules over a Ring ............................................... 84
4.3 Semisimple Modules............................................................................. 91
4.4 Matrix Rings ........................ ................................ ................. ................. 96
4.5 Direct Products of Rings...................................................................... 101
4.6 Free Modules ......................................................................................... 105
4.7 Projective and Injective Modules ...................... .................................. 110
4.8 The Tensor Product of Modules ......................................................... 117
4.9 Duality of Finite Abelian Groups........................................................ 125
v
vi Basic Algebra
5. Algebras
5.1 Algebras; Definition and Examples ..................................................... 131
5.2 The Wedderburn Structure Theorems................................................ 137
5.3 The Radical............................................................................................ 141
5.4 The Tensor Product of Algebras ......................................................... 146
5.5 The Regular Representation; Norm and Trace.................................. 153
5.6 Mobius Functions ................................................................................. 157
6. Multilinear Algebra
6.1 Graded Algebras .................................................................................... 165
6.2 Free Algebras and Tensor Algebras..................................................... 168
6.3 The Hilbert Series of a Graded Ring or Module............................... 173
6.4 The Exterior Algebra on a Module ..................................................... 179
7. Field Theory
7.1 Fields and their Extensions.................................................................. 189
7.2 Splitting Fields....................................................................................... 195
7.3 The Algebraic Closure of a Field......................................................... 200
7.4 Separability............................................................................................. 203
7.5 Automorphisms of Field Extensions................................................... 206
7.6 The Fundamental Theorem of Galois Theory ................................... 211
7.7 Roots of Unity ....................................................................................... 217
7.8 Finite Fields............................................................................................ 223
7.9 Primitive Elements; Norm and Trace................................................. 227
7.10 Galois Theory of Equations ................................................................. 232
7.11 The Solution of Equations by Radicals .............................................. 238
8. Quadratic Forms and Ordered Fields
8.1 Inner Product Spaces ....................................................... ................ ..... 249
8.2 Orthogonal Sums and Diagonalization .............................................. 252
8.3 The Orthogonal Group of a Space................................. ..................... 256
8.4 The Clifford Algebra and the Spinor Norm ...................................... 259
8.5 Witt's Cancellation Theorem and the Witt Group of a Field ......... 268
8.6 Ordered Fields ....................................................................................... 272
8.7 The Field of Real Numbers.................................................................. 275
8.8 Formally Real Fields.............................................................................. 279
8.9 The Witt Ring of a Field...................................................................... 291
8.10 The Symplectic Group.......................................................................... 298
8.11 Quadratic Forms in Characteristic Two............................................. 301
9. Valuation Theory
9.1 Divisibility and Valuations................................................................... 307
9.2 Absolute Values..................................................................................... 312
9.3 The p-adic Numbers............................................................................. 322
9.4 Integral Elements................................................................................... 331
9.5 Extension of Valuations ....................................................................... 336
Contents vii
10. Commutative Rings
10.1 Operations on Ideals............................................................................. 347
10.2 Prime Ideals and Factorization............................................................ 349
10.3 Localization............................................................................................ 354
lOA Noetherian Rings................................................................................... 361
10.5 Dedekind Domains ............................................................................... 362
10.6 Modules over Dedekind Domains ...................................................... 371
10.7 Algebraic Equations .............................................................................. 376
10.8 The Primary Decomposition ............................................................... 380
10.9 Dimension.............................................................................................. 386
10.10 The Hilbert Nullstellensatz................................................................... 391
11. Infinite Field Extensions
11.1 Abstract Dependence Relations ........................................................... 397
11.2 Algebraic Dependence........................................................................... 402
11.3 Simple Transcendental Extensions ...................................................... 405
11.4 Separable and p-radical Extensions ..................................................... 409
11.5 Derivations ................ ............................................................................. 414
11.6 Linearly Disjoint Extensions ......................................... ....................... 418
11.7 Composites of Fields............................................................................. 427
11.8 Infinite Algebraic Extensions ............................................................... 431
11.9 Galois Descent ....................................................................................... 437
11.10 Kummer Extensions.............................................................................. 441
Bibliography .............................................................................................................. 449
List of Notations ...................................................................................................... 453
Author Index ............................................................................................................ 457
Subject Index ................................................................ ............................................ 459

Preface
Much of the second and third year undergraduate course in mathematics (as well as
some graduate work) was covered by Volumes 2 and 3 of my book on algebra, now
out of print. 1 So I was very pleased when Springer Verlag offered to bring out a new
version of these volumes. The present book is based on both these volumes, comple
mented by the definitions and basic facts on groups and rings. Thus the volume is
addressed to students who have some knowledge of linear algebra and who have
met groups and fields, though all the essential facts are recalled here. My overall
aim has been to present as many of the important results in algebra as would
conveniently fit into one volume. It is my hope to collect the remaining parts of
Volumes 2 and 3 into a second book, more oriented towards applications.2
Apart from chapters on groups (Chapter 2), rings and modules (Chapters 4, 5 and 6)
and fields (Chapters 7 and 11), a number of concepts are treated that are less central
but nevertheless have many uses. Chapter 1, on set theory, deals with countable and
well-ordered sets, as well as Zorn's lemma and a brief section on graphs. Chapter 3
introduces lattices and categories, both concepts that form an important part of
the language of modern algebra. The general theory of quadratic forms has many
links with ordered fields, which are developed in Chapter 8. Chapters 9 and 10 are
devoted to valuation theory and commutative rings, a subject that has gained in
importance through its use in algebraic geometry.
On a first encounter some readers may find the style of this book somewhat
concise, but they should bear in mind that mathematical texts are best read with
paper and pencil, to work out the full consequences of what is being said and to
check examples. The matter has been well put by Einstein, who said: "Everything
should be explained as far as possible but no further." There are numerous exercises
throughout, with occasional hints (but no solutions), and some historical remarks.
My thanks are due to the staff of Springer Verlag for the efficient way they have
produced this volume.
University College London P.M. Cohn
June 2002
I Algebra, Vol. 2 (2nd edn, 1989) and Vol. 3 (2nd edn, 1991), Wiley and Sons.
2 Further Algebra and Applications, Springer Verlag, London (2003). Referred to in the text as FA.
ix