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ALGEBRA AND NUMBER THEORY
ALGEBRA AND NUMBER THEORY
An Integrated Approach
MARTYN R. DIXON
Department of Mathematics
University of Alabama
LEONID A. KURDACHENKO
Department of Algebra
School of Mathematics and Mechanics
National University of Dnepropetrovsk
IGOR VA. SUBBOTIN
Department of Mathematics and Natural Sciences
National University
~WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Dixon, Martyn R. (Martyn Russell), 1955-
Algebra and number theory : an integrated approach I Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin.
p. em. Includes index.
ISBN 978-0-470-49636-7 (cloth)
I. Number theory. 2. Algebra. I. Kurdachenko, L. II. Subbotin, Igor Ya., 1950III. Title. QA24l.D59 2010
512-dc22
2010003428
Printed in Singapore
10987654321
CONTENTS
PREFACE |
ix |
CHAPTER 1 SETS |
1 |
1.1Operations on Sets I
Exercise Set 1.1 I 6
1.2Set Mappings I 8 Exercise Set 1.2 I 19
1.3Products of Mappings I 20 Exercise Set 1.3 I 26
1.4Some Properties of Integers I 28 Exercise Set 1.4 I 39
CHAPTER 2 MATRICES AND DETERMINANTS |
41 |
2.1Operations on Matrices I 41 Exercise Set 2.1 I 52
2.2Permutations of Finite Sets I 54 Exercise Set 2.2 I 64
2.3Determinants of Matrices I 66 Exercise Set 2.3 I 77
2.4Computing Determinants I 79 Exercise Set 2.4 I 91
v
vi CONTENTS
2.5Properties of the Product of Matrices I 93 Exercise Set 2.5 I 103
CHAPTER 3 FIELDS |
105 |
3.1Binary Algebraic Operations I 105 Exercise Set 3.1 I 118
3.2Basic Properties of Fields I 119 Exercise Set 3.2 I 129
3.3The Field of Complex Numbers I 130 Exercise Set 3.3 I 144
CHAPTER4 VECTOR SPACES |
145 |
4.1Vector Spaces I 146 Exercise Set 4.1 I 158
4.2Dimension I 159 Exercise Set 4.2 I 172
4.3The Rank of a Matrix I 174 Exercise Set 4.3 I 181
4.4Quotient Spaces I 182 Exercise Set 4.4 I 186
CHAPTER 5 LINEAR MAPPINGS |
187 |
5.1Linear Mappings I 187 Exercise Set 5.1 I 199
5.2Matrices of Linear Mappings I 200 Exercise Set 5.2 I 207
5.3Systems of Linear Equations I 209 Exercise Set 5.3 I 215
5.4Eigenvectors and Eigenvalues I 217 Exercise Set 5.4 I 223
CHAPTER 6 BILINEAR FORMS |
226 |
6.1Bilinear Forms I 226 Exercise Set 6.1 I 234
6.2Classical Forms I 235 Exercise Set 6.2 I 247
6.3Symmetric Forms over ffi. I 250 Exercise Set 6.3 I 257
CONTENTS vii
6.4Euclidean Spaces I 259 Exercise Set 6.4 I 269
CHAPTER 7 RINGS |
272 |
7.1Rings, Subrings, and Examples I 272 Exercise Set 7.1 I 287
7.2Equivalence Relations I 288 Exercise Set 7.2 I 295
7.3Ideals and Quotient Rings I 297 Exercise Set 7.3 I 303
7.4Homomorphisms of Rings I 303 Exercise Set 7.4 I 313
7.5Rings of Polynomials and Formal Power Series I 315
Exercise Set 7.5 I 327
7.6Rings of Multivariable Polynomials I 328 Exercise Set 7.6 I 336
CHAPTERS GROUPS |
338 |
8.1Groups and Subgroups I 338 Exercise Set 8.1 I 348
8.2Examples of Groups and Subgroups I 349 Exercise Set 8.2 I 358
8.3Cosets I 359
Exercise Set 8.3 I 364
8.4Normal Subgroups and Factor Groups I 365 Exercise Set 8.4 I 374
8.5Homomorphisms of Groups I 375 Exercise Set 8.5 I 382
CHAPTER9 ARITHMETIC PROPERTIES OF RINGS |
384 |
9.1Extending Arithmetic to Commutative Rings I 384 Exercise Set 9.1 I 399
9.2Euclidean Rings I 400 Exercise Set 9.2 I 404
9.3Irreducible Polynomials I 406 Exercise Set 9.3 I 415
Viii CONTENTS
9.4Arithmetic Functions I 416 Exercise Set 9.4 I 429
9.5Congruences I 430 Exercise Set 9.5 I 446
CHAPTER 10 THE REAL NUMBER SYSTEM |
448 |
10.1The Natural Numbers I 448
10.2The Integers I 458
10.3The Rationals I 468
10.4The Real Numbers I 477
ANSWERS TO SELECTED EXERCISES |
489 |
INDEX |
513 |
PREFACE
Algebra and number theory are two powerful, established branches of modem mathematics at the forefront of current mathematical research which are playing an increasingly significant role in different branches of mathematics (for instance, in geometry, topology, differential equations, mathematical physics, and others) and in many relatively new applications of mathematics such as computing, communications, and cryptography. Algebra also plays a role in many applications of mathematics in diverse areas such as modem physics, crystallography, quantum mechanics, space sciences, and economic sciences.
Preface Historically, algebra and number theory have developed together, enriching each other in the process, and this often makes it difficult to draw a precise boundary separating these subjects. It is perhaps appropriate to say that they actually form one common subject: algebra and number theory. Thus, results in number theory are the basis and "a type of sandbox" for algebraic ideas and, in tum, algebraic tools contribute tremendously to number theory. It is interesting to note that newly developed branches of mathematics such as coding theory heavily use ideas and results from both linear algebra and number theory.
There are three mandatory courses, linear algebra, abstract algebra, and number theory, in all university mathematics programs that every student of mathematics should take. Increasingly, it is also becoming evident that students of computer science and other such disciplines also need a strong background in these three areas. Most of the time, these three disciplines are the subject of different and separate lecture courses that use different books dedicated to each subject individually. In a curriculum that is increasingly stretched by the need to offer traditional favorites, while introducing new applications, we think that it is desirable to introduce a fresh approach to the way these three specific courses are taught. On the basis of the authors' experience, we think that one course, integrating these three
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