Basic Abstract Algebra by R.B. Bhattacharya

Basic Abstract Algebra
 
Author:
R.B. Bhattacharya, S.K.Jain & & P. Nagpal
Publisher: Cambridge University Press
ISBN No: 0-521-46629-6
Release at: 1994
Pages: 277
Edition:
Second Edition
File Size: 8 MB
File Type: pdf
Language: English



Content of Basic Abstract Algebra



Part I Preliminaries


Chapter I Sets and mappings 3

1. Sets 3

2. Relations 9

3. Mappings 14

4. Binary operations 21

5. Cardinality of a set 25

Chapter 2 Integers, real numbers, and complex numbers 30

I. Integers 30

2. Rational, real, and complex numbers 35

3. Fields 36

Chapter 3 Matrices and determinants 39

1. Matrices 39

2. Operations on matrices 41

3. Partitions of a matrix 46

4. The determinant function 47

5. Properties of the determinant function 49

6. Expansion of det A 53

Part 11 Groups

Chapter 4 Groups 61

1. Semigroups and groups 61

2. Homomorphisms 69

3. Subgroups and cosets 72

4. Cyclic groups 82

5. Permutation groups 84

6. Generators and relations 90

Chapter 5 Normal subgroups 91

1. Normal subgroups and quotient groups 91

2. Isomorphism theorems 97

3. Automorphisms 104

4. Conjugacy and G-sets 107

Chapter 6 Normal series 120

1. Normal series 120

2. Solvable groups 124

3. Nilpotent groups 126

Chapter 7 Permutation groups 129

1. Cyclic decomposition 129

2. Alternating group 132

3. Simplicity of 135

Chapter 8 Structure theorems of groups 138

1. Direct products 138

2. Finitely generated abelian groups 141

3. Invariants of a finite abelian group 143

4. Sylow theorems 146

5. Groups of orders p2. PQ 152


Part III Rings and modules


Chapter 9 Rings 159

1. Definition and examples 159

2. Elementary properties of rings 161

3. Types of rings 163

4. Subrings and characteristic of a ring 168

5. Additional examples of rings 176

Chapter 10 Ideals and homomorphisms 179

1. Ideals 179

2. Homomorphisms 187

3. Sum and direct sum of ideals 196

4. Maximal and prime ideals 203

5. Nilpotent and nil ideals 209

6. Zorn's lemma 210

Chapter II Unique factorization domains and euclidean

domains 212

1. Unique factorization domains 212

2. Principal ideal domains 216

3. Euclidean domains 217

4. Polynomial rings over UFD 219

Chapter 12 Rings of fractions 224

1. Rings of fractions 224

2. Rings with Ore condition 228

Chapter 13 Integers 233

1, Peano's axioms 233

2. Integers 240

Chapter 14 Modules and vector spaces 246

1. Definition and examples 246

2. Submodules and direct sums 248

3. R-homomorphisms and quotient modules 253

4. Completely reducible modules 260

5. Free modules 263

6. Representation of linear mappings 268

7. Rank of a linear mapping 273


Part IV Field theory


Chapter 15 Algebraic extensions of fields 281

1. Irreducible polynomials and Eisenstein criterion 281

2. Adjunction of roots 285

3. Algebraic extensions 289

4. Algebraically closed fields 295

Chapter 16 Normal and separable extensions 300

1. Splitting fields 300

2. Normal extensions 304

3. Multiple roots 307

4. Finite fields 310

5. Separable extensions 316

Chapter 17 Galois theory 322

1. Automorphism groups and fixed fields 322

2. Fundamental theorem of Galois theory 330

3. Fundamental theorem of algebra 338

Chapter 18 Applications of Galois theory to classical problems 340

1. Roots of unity and cyclotomic polynomials 340

2. Cyclic extensions 344

3. Polynomials solvable by radicals 348

4. Symmetric functions 355

5. Ruler and compass constructions 358


Part V Additional topics


Chapter 19 Noetherian and artinian modules and rings 367

1. HomR 367

2. Noetherian and artinian modules 368

3. Wedderburn—Artin theorem 382

4. Uniform modules, primary modules, and Noether—Lasker theorem 388

Chapter 20 Smith normal form over a PID and rank 392

1. Preliminaries 392

2. Row module, column module, and rank 393

3. Smith normal form 394

Chapter 21 Finitely generated modules over a PID 402

1. Decomposition theorem 402

2. Uniqueness of the decomposition 404

3. Application to finitely generated abelian groups 408

4. Rational canonical form 409

5. Generalized Jordan form over any field 418

Chapter 22 Tensor products 426

1. Categories and functors 426

2. Tensor products 428

3. Module structure of tensor product 431

4. Tensor product of homomorphisms 433

5. Tensor product of algebras 436

Solutions to odd-numbered problems 438

Selected bibliography 476

Index 477

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