Algebra : Abstract and Modern.

Format:

Book

Author/Creator:

Swamy, U. M.

Contributor:

Murthy, A. V. S. N.

Language:

English

Subjects (All):

Algebra.

Physical Description:

1 online resource (776 pages)

Edition:

0

Other Title:

Algebra

Place of Publication:

Noida : Pearson India, 2012.

Summary:

Algebra: Abstract and Modern, spread across 16 chapters, introduces the reader to the preliminaries of algebra and then explains topics like group theory and field theory in depth. It also features a blend of numerous challenging exercises and examples that further enhance each chapter. Covering all the necessary topics on the subject, this text is an ideal text book for an undergraduate course on mathematics.

Contents:

Cover

Contents

Preface

Part I: Preliminaries

Chapter 1: Sets and Relations

1.1 Sets and subsets

1.2 Relations and functions

1.3 Equivalence relations and partitions

1.4 The cardinality of a set

Chapter 2: Number Systems

2.1 Integers

2.2 Congruence modulo n

2.3 Rational, real and complex numbers

2.4 Ordering

2.5 Matrices

2.6 Determinants

Part II: Group Theory

Chapter 3: Groups

3.1 Binary systems

3.2 Groups

3.3 Elementary properties of groups

3.4 Finite groups and group tables

Chapter 4: Subgroups and Quotient Groups

4.1 Subgroups

4.2 Cyclic groups

4.3 Cosets of a subgroup

4.4 Lagrange's theorem

4.5 Normal subgroups

4.6 Quotient groups

Chapter 5: Homomorphisms of Groups

5.1 Definition and examples

5.2 Fundamental theorem of homomorphisms

5.3 Isomorphism theorems

5.4 Automorphisms

Chapter 6: Permutation Groups

6.1 Cayley's theorem

6.2 The symmetric group Sn

6.3 Cycles

6.4 Alternating group An and dihedral group Dn

Chapter 7: Group Actions on Sets

7.1 Action of a group on a set

7.2 Orbits and stabilizers

7.3 Certain counting techniques

7.4 Cauchy and Sylow theorems

Chapter 8: Structure Theory of Groups

8.1 Direct products

8.2 Finitely generated abelian groups

8.3 Invariants of finite abelian groups

8.4 Groups of small order

Part III: Ring Theory

Chapter 9: Rings

9.1 Examples and elementary properties

9.2 Certain special elements in rings

9.3 The characteristic of a ring

9.4 Subrings

9.5 Homomorphisms of rings

9.6 Certain special types of rings

9.7 Integral domains and fields

Chapter 10: Ideals and Quotient Rings

10.1 Ideals

10.2 Quotient rings

10.3 Chinese remainder theorem

10.4 Prime ideals

10.5 Maximal ideals

10.6 Embeddings of rings.

Chapter 11: Polynomial Rings

11.1 Rings of polynomials

11.2 The division algorithm

11.3 Polynomials over a field

11.4 Irreducible polynomials

Chapter 12: Factorization in Integral Domains

12.1 Divisibility in integral domains

12.2 Principal ideal domains

12.3 Unique factorization domains

12.4 Polynomials over UFDs

12.5 Euclidean domains

12.6 Some applications to number theory

Chapter 13: Modules and Vector Spaces

13.1 Modules and submodules

13.2 Homomorphisms and quotients of modules

13.3 Direct products and sums

13.4 Simple and completely reducible modules

13.5 Free modules

13.6 Vector spaces

Part IV: Field Theory

Chapter 14: Extension Fields

14.1 Extensions of a field

14.2 Algebraic extensions

14.3 Algebraically closed fields

14.4 Derivatives and multiple roots

14.5 Finite fields

Chapter 15: Galois Theory

15.1 Separable and normal extensions

15.2 Automorphism groups and fixed fields

15.3 Fundamental theorem of Galois theory

Chapter 16: Selected Applications of Galois Theory

16.1 Fundamental theorem of algebra

16.2 Cyclic extensions

16.3 Solvable groups

16.4 Polynomials solvable by radicals

16.5 Constructions by ruler and compass

Answers/Hints to Selected Even-Numbered Exercises

Index.

Notes:

Description based on publisher supplied metadata and other sources.

Includes index.

ISBN:

93-325-0993-X

OCLC:

1024272395