Algebra abstract and modern

776 Pages·2012·40.154 MB·English

by  Murty, A. V. S. N., Swamy, U. M

Algebra – Abstract
and Modern
Dr U. M. Swamy
Dean, Faculty of Science (Retired)
Department of Mathematics
Andhra University
Visakhapatnam
Dr A. V. S. N. Murty
Professor of Mathematics
Srinivasa Institute of Engineering and Technology
Amalapuram
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Copyright © 2012 Dorling Kindersley (India) Pvt. Ltd
Licensees of Pearson Education in South Asia
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without the publisher’s prior written consent.
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ISBN 9788131758922
eISBN 9789332509931
Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA
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Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi
110 017, India
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To my grandfather – Late Sri. Akella Srihari,
To my father – Late Sri. Akella Krishna Murty,
To my teacher – Prof. U. M. Swamy
and
My family members
A. V. S. N. Murty
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Contents
Preface ix
Part I: Preliminaries
1. Sets and Relations 1-3
1.1 Sets and subsets 1-3
1.2 Relations and functions 1-11
1.3 Equivalence relations and partitions 1-21
1.4 The cardinality of a set 1-27
2. Number Systems 2-1
2.1 Integers 2-1
2.2 Congruence modulo n 2-13
2.3 Rational, real and complex numbers 2-23
2.4 Ordering 2-30
2.5 Matrices 2-34
2.6 Determinants 2-43
Part II: Group Theory
3. Groups 3-3
3.1 Binary systems 3-3
3.2 Groups 3-16
3.3 Elementary properties of groups 3-32
3.4 Finite groups and group tables 3-45
4. Subgroups and Quotient Groups 4-1
4.1 Subgroups 4-2
4.2 Cyclic groups 4-12
4.3 Cosets of a subgroup 4-24
4.4 Lagrange’s theorem 4-30
4.5 Normal subgroups 4-39
4.6 Quotient groups 4-45
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vi Contents
5. Homomorphisms of Groups 5-1
5.1 Definition and examples 5-2
5.2 Fundamental theorem of homomorphisms 5-16
5.3 Isomorphism theorems 5-23
5.4 Automorphisms 5-29
6. Permutation Groups 6-1
6.1 Cayley’s theorem 6-1
6.2 The symmetric group S 6-7
n
6.3 Cycles 6-11
6.4 Alternating group A and dihedral group D 6-23
n n
7. Group Actions on Sets 7-1
7.1 Action of a group on a set 7-1
7.2 Orbits and stabilizers 7-8
7.3 Certain counting techniques 7-19
7.4 Cauchy and Sylow theorems 7-28
8. Structure Theory of Groups 8-1
8.1 Direct products 8-1
8.2 Finitely generated abelian groups 8-12
8.3 Invariants of finite abelian groups 8-29
8.4 Groups of small order 8-33
Part III: Ring Theory
9. Rings 9-3
9.1 Examples and elementary properties 9-3
9.2 Certain special elements in rings 9-16
9.3 The characteristic of a ring 9-22
9.4 Subrings 9-25
9.5 Homomorphisms of rings 9-29
9.6 Certain special types of rings 9-35
9.7 Integral domains and fields 9-43
10. Ideals and Quotient Rings 10-1
10.1 Ideals 10-1
10.2 Quotient rings 10-20
10.3 Chinese remainder theorem 10-29
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Contents vii
10.4 Prime ideals 10-34
10.5 Maximal ideals 10-43
10.6 Embeddings of rings 10-56
11. Polynomial Rings 11-1
11.1 Rings of polynomials 11-1
11.2 The division algorithm 11-15
11.3 Polynomials over a field 11-25
11.4 Irreducible polynomials 11-31
12. Factorization in Integral Domains 12-1
12.1 Divisibility in integral domains 12-2
12.2 Principal ideal domains 12-10
12.3 Unique factorization domains 12-18
12.4 Polynomials over UFDs 12-26
12.5 Euclidean domains 12-36
12.6 Some applications to number theory 12-44
13. Modules and Vector Spaces 13-1
13.1 Modules and submodules 13-2
13.2 Homomorphisms and quotients of modules 13-10
13.3 Direct products and sums 13-18
13.4 Simple and completely reducible modules 13-31
13.5 Free modules 13-35
13.6 Vector spaces 13-42
Part IV: Field Theory
14. Extension Fields 14-3
14.1 Extensions of a field 14-3
14.2 Algebraic extensions 14-8
14.3 Algebraically closed fields 14-20
14.4 Derivatives and multiple roots 14-27
14.5 Finite fields 14-33
15. Galois Theory 15-1
15.1 Separable and normal extensions 15-1
15.2 Automorphism groups and fixed fields 15-10
15.3 Fundamental theorem of Galois theory 15-19
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viii Contents
16. Selected Applications of Galois Theory 16-1
16.1 Fundamental theorem of algebra 16-2
16.2 Cyclic extensions 16-5
16.3 Solvable groups 16-8
16.4 Polynomials solvable by radicals 16-11
16.5 Constructions by ruler and compass 16-20
Answers/Hints to Selected Even-Numbered Exercises A-1
Index I-1
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Preface
This book is designed for a two-semester sequence as a first course in abstract
algebra for advanced undergraduate and junior post-graduate students. A
glance at the table of contents will reveal the scope of the book; the range of
topics covered is reasonably standard, with no major surprises. Our intention
is to present a text that is logically developed, precise, and in keeping with
the spirit of the times. Guided by the principle that a routine diet of defini-
tions, theorems and results soon becomes unpalatable, we have concentrated
on supplementing the concepts with examples and counter-examples and on
establishing the important and fruitful results in a formal, rigorous fashion.
En route, we have tried to showcase the power and elegance of the abstract –
modern approach in mathematics, particularly in algebra, and chosen the title
‘Algebra – Abstract and Modern’ for this book.
The reader is not presumed to possess any previous knowledge of the con-
cepts of modern algebra, except certain mathematical maturity and a will to
learn abstract thinking. Consequently, the book’s initial chapters are some-
what elementary, with the exposition proceeding at a leisurely pace, filling in
the details of proofs, particularly of basic results. To smoothen the approach,
we have devoted Part I to preliminaries consisting of two chapters, one on
sets, relations, function, partitions and the cardinality of a set and the other
on number systems, matrices and determinants. This part also serves as a
vehicle for introducing some of the notation and terminology concerning the
language of basic mathematics to be used in the later parts. Proofs of most of
the results in Part I are skipped and given as exercises to encourage interested
readers to work on them.
There are three parts in the main text of the book, Part II (Chapter 3–8),
Part III (Chapter 9–13) and Part IV (Chapters 14–16) covering Group
Theory, Ring Theory and Field Theory respectively. Each chapter is divided
into a suitable number of sections in which definitions of the various con-
cepts are immediately followed by a sufficient number of examples and
counter-examples. Worked exercises are included in each section in addition
to a set of exercises of varying levels of difficulty at the end of each section.
These exercises are an integral part of the book and require the reader’s
active participation. Some of them introduce a variety of ideas not treated
in the body of the text and impart certain additional information about con-
cepts discussed in chapters. We have given a brief introduction of vector
spaces and linear transformations to the extent necessary for a discussion
on Galois Theory. We have resisted the temptation to use Exercises, except
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