Basic Abstract Algebra (complete ver)

Description: This book will get you there if you believe in it. It has examples with solutions and problems with solutions. The only topic that does not have problems with solutions is categories. For this, I have the Hungerford text, and I am presently in the process of finding a better book for this. Otherwise it is the perfect book for self-study. In addition to many new problems for practice and challenge, this edition of a self-contained graduate text on abstract algebra contains an introduction to lattices, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Lasker-Noether theorem. P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul - Basic Abstract Algebra, 2nd Edition-Cambridge University Press (1994) 1 Front Cover 1 Title 4 Copyright 5 Dedication 6 Contents 8 Preface to the second edition 14 Preface to the first edition 15 Glossary of symbols 19 Part I Preliminaries 22 Chapter I Sets and mappings 24 1. Sets 24 2. Relations 30 3. Mappings 35 4. Binary operations 42 5. Cardinality of a set 46 Chapter 2 Integers, real numbers, and complex numbers 51 1. Integers 51 2. Rational, real, and complex numbers 56 3. Fields 57 Chapter 3 Matrices and determinants 60 1. Matrices 60 2. Operations on matrices 62 3. Partitions of a matrix 67 4. The determinant function 68 5. Properties of the determinant function 70 6. Expansion of det A 74 Part II Groups 80 Chapter 4 Groups 82 I. Semigroups and groups 82 2. Homomorphisms 90 3. Subgroups and cosets 93 4. Cyclic groups 103 5. Permutation groups 105 6. Generators and relations 111 Chapter 5 Normal subgroups 112 1. Normal subgroups and quotient groups 112 2. Isomorphism theorems 118 3. Automorphisms 125 4. Conjugacy and G-sets 128 Chapter 6 Normal series 141 1. Normal series 141 2. Solvable groups 145 3. Nilpotent groups 147 Chapter 7 Permutation groups 150 1. Cyclic decomposition 150 2. Alternating group 153 3. Simplicity of 156 Chapter 8 Structure theorems of groups 159 1. Direct products 159 2. Finitely generated abelian groups 162 3. Invariants of a finite abelian group 164 4. Sylow theorems 167 5. Groups of orders p2. pq 173 Part III Rings and modules 178 Chapter 9 Rings 180 1. Definition and examples 180 2. Elementary properties of rings 182 3. Types of rings 184 4. Subrings and characteristic of a ring 189 5. Additional examples of rings 197 Chapter 10 Ideals and homomorphisms 200 1. Ideals 200 2. Homomorphisms 208 3. Sum and direct sum of ideals 217 4. Maximal and prime ideals 224 5. Nilpotent and nil ideals 230 6. Zorn's lemma 231 Chapter 11 Unique factorization domains and euclidean domains 233 1. Unique factorization domains 233 2. Principal ideal domains 237 3. Euclidean domains 238 4. Polynomial rings over UFD 240 Chapter 12 Rings of fractions 245 1. Rings of fractions 245 2. Rings with Ore condition 249 Chapter 13 Integers 254 1, Peano's axioms 254 2. Integers 261 Chapter 14 Modules and vector spaces 267 1. Definition and examples 267 2. Submodules and direct sums 269 3. R-homomorphisms and quotient modules 274 4. Completely reducible modules 281 5. Free modules 284 6. Representation of linear mappings 289 7. Rank of a linear mapping 294 Chapter 15 Algebraic extensions of fields 302 1. Irreducible polynomials and Eisenstein criterion 302 2. Adjunction of roots 306 3. Algebraic extensions 310 4. Algebraically closed fields 316 Chapter 16 Normal and separable extensions 321 1. Splitting fields 321 2. Normal extensions 325 3. Multiple roots 328 4. Finite fields 331 5. Separable extensions 337 Chapter 17 Galois theory 343 1. Automorphism groups and fixed fields 343 2. Fundamental theorem of Galois theory 351 3. Fundamental theorem of algebra 359 Chapter 18 Applications of Galois theory to classical problems 361 1. Roots of unity and cyclotomic polynomials 361 2. Cyclic extensions 365 3. Polynomials solvable by radicals 369 4. Symmetric functions 376 5. Ruler and compass constructions 379 Chapter 19 Noetherian and artinian modules and rings 388 1. HomR 388 2. Noetherian and artinian modules 389 3. Wedderburn?rtin theorem 403 4. Uniform modules, primary modules, and Noether?asker theorem 409 Chapter 20 Smith normal form over a PID and rank 413 1. Preliminaries 413 2. Row module, column module, and rank 414 3. Smith normal form 415 Chapter 21 Finitely generated modules over a PID 423 1. Decomposition theorem 423 2. Uniqueness of the decomposition 425 3. Application to finitely generated abelian groups 429 4. Rational canonical form 430 5. Generalized Jordan form over any field 439 Chapter 22 Tensor products 447 1. Categories and functors 447 2. Tensor products 449 3. Module structure of tensor product 452 4. Tensor product of homomorphisms 454 5. Tensor product of algebras 457 Solutions to odd-numbered problems 459 Selected bibliography 497 Index 498 Back Cover 509 Front Cover......Page 1 Title......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Preface to the second edition......Page 14 Preface to the first edition......Page 15 Glossary of symbols......Page 19 Part I Preliminaries......Page 22 1. Sets ......Page 24 2. Relations ......Page 30 3. Mappings ......Page 35 4. Binary operations ......Page 42 5. Cardinality of a set......Page 46 1. Integers ......Page 51 2. Rational, real, and complex numbers ......Page 56 3. Fields ......Page 57 1. Matrices ......Page 60 2. Operations on matrices ......Page 62 3. Partitions of a matrix ......Page 67 4. The determinant function ......Page 68 5. Properties of the determinant function ......Page 70 6. Expansion of det A ......Page 74 Part II Groups......Page 80 I. Semigroups and groups......Page 82 2. Homomorphisms ......Page 90 3. Subgroups and cosets ......Page 93 4. Cyclic groups ......Page 103 5. Permutation groups ......Page 105 6. Generators and relations ......Page 111 1. Normal subgroups and quotient groups ......Page 112 2. Isomorphism theorems ......Page 118 3. Automorphisms ......Page 125 4. Conjugacy and G-sets......Page 128 1. Normal series ......Page 141 2. Solvable groups ......Page 145 3. Nilpotent groups ......Page 147 1. Cyclic decomposition......Page 150 2. Alternating group ......Page 153 3. Simplicity of ......Page 156 1. Direct products ......Page 159 2. Finitely generated abelian groups ......Page 162 3. Invariants of a finite abelian group ......Page 164 4. Sylow theorems ......Page 167 5. Groups of orders p2. pq ......Page 173 Part III Rings and modules......Page 178 1. Definition and examples ......Page 180 2. Elementary properties of rings......Page 182 3. Types of rings ......Page 184 4. Subrings and characteristic of a ring ......Page 189 5. Additional examples of rings ......Page 197 1. Ideals ......Page 200 2. Homomorphisms ......Page 208 3. Sum and direct sum of ideals ......Page 217 4. Maximal and prime ideals ......Page 224 5. Nilpotent and nil ideals ......Page 230 6. Zorn's lemma ......Page 231 1. Unique factorization domains ......Page 233 2. Principal ideal domains ......Page 237 3. Euclidean domains......Page 238 4. Polynomial rings over UFD ......Page 240 1. Rings of fractions ......Page 245 2. Rings with Ore condition ......Page 249 1, Peano's axioms ......Page 254 2. Integers ......Page 261 1. Definition and examples ......Page 267 2. Submodules and direct sums ......Page 269 3. R-homomorphisms and quotient modules ......Page 274 4. Completely reducible modules ......Page 281 5. Free modules ......Page 284 6. Representation of linear mappings ......Page 289 7. Rank of a linear mapping ......Page 294 1. Irreducible polynomials and Eisenstein criterion ......Page 302 2. Adjunction of roots ......Page 306 3. Algebraic extensions ......Page 310 4. Algebraically closed fields......Page 316 1. Splitting fields ......Page 321 2. Normal extensions......Page 325 3. Multiple roots ......Page 328 4. Finite fields ......Page 331 5. Separable extensions ......Page 337 1. Automorphism groups and fixed fields ......Page 343 2. Fundamental theorem of Galois theory ......Page 351 3. Fundamental theorem of algebra ......Page 359 1. Roots of unity and cyclotomic polynomials ......Page 361 2. Cyclic extensions......Page 365 3. Polynomials solvable by radicals ......Page 369 4. Symmetric functions ......Page 376 5. Ruler and compass constructions ......Page 379 1. HomR ......Page 388 2. Noetherian and artinian modules ......Page 389 3. Wedderburn?rtin theorem ......Page 403 4. Uniform modules, primary modules, and Noether?asker theorem ......Page 409 1. Preliminaries ......Page 413 2. Row module, column module, and rank ......Page 414 3. Smith normal form......Page 415 1. Decomposition theorem ......Page 423 2. Uniqueness of the decomposition ......Page 425 3. Application to finitely generated abelian groups ......Page 429 4. Rational canonical form ......Page 430 5. Generalized Jordan form over any field ......Page 439 1. Categories and functors ......Page 447 2. Tensor products ......Page 449 3. Module structure of tensor product ......Page 452 4. Tensor product of homomorphisms ......Page 454 5. Tensor product of algebras......Page 457 Solutions to odd-numbered problems ......Page 459 Selected bibliography......Page 497 Index ......Page 498 Back Cover......Page 509 This Book Represents A Complete Course In Abstract Algebra, Providing Instructors With Flexibility In The Selection Of Topics To Be Taught In Individual Classes. All The Topics Presented Are Discussed In A Direct And Detailed Manner. Throughout The Text, Complete Proofs Have Been Given For All Theorems Without Glossing Over Significant Details Or Leaving Important Theorems As Exercises. The Book Contains Many Examples Fully Worked Out And A Variety Of Problems For Practice And Challenge. Solutions To The Odd-numbered Problems Are Provided At The End Of The Book. This New Edition Contains An Introduction To Lattices, A New Chapter On Tensor Products And A Discussion Of The New (1993) Approach To The Celebrated Lasker–noether Theorem. In Addition, There Are Over 100 New Problems And Examples, Particularly Aimed At Relating Abstract Concepts To Concrete Situations. Pt. I. Preliminaries -- Sets And Mappings -- Integers, Real Numbers, And Complex Numbers -- Matrices And Determinants -- Pt. Ii. Groups -- Groups -- Normal Subgroups -- Normal Series -- Permutation Groups -- Structure Theorems Of Groups -- Pt. Iii. Rings And Modules -- Rings -- Ideals And Homomorphisms -- Unique Factorization Domains And Euclidean Domains -- Rings Of Fractions -- Integers -- Modules And Vector Spaces -- Pt. Iv. Field Theory -- Algebraic Extensions Of Fields -- Normal And Separable Extensions -- Galois Theory -- Applications Of Galios Theory To Classical Problems -- Pt. V. Additional Topics -- Noetherian And Artinian Modules And Rings -- Smith Normal Form Over A Pid And Rank -- Finitely Generated Modules Over A Pid -- Tensor Products P.b. Bhattacharya, S.k. Jain, S.r. Nagpaul. Includes Bibliographical References (p. 476) And Index. This is a self-contained text on abstract algebra for senior undergraduate and senior graduate students, which gives complete and comprehensive coverage of the topics usually taught at this level. The book is divided into five parts. The first part contains fundamental information such as an informal introduction to sets, number systems, matrices, and determinants. The second part deals with groups. The third part treats rings and modules. The fourth part is concerned with field theory. Much of the material in parts II, III, and IV forms the core syllabus of a course in abstract algebra. The fifth part goes on to treat some additional topics not usually taught at the undergraduate level, such as the Wedderburn-Artin theorem for semisimple artinian rings, Noether-Lasker theorem, the Smith-Normal form over a PID, finitely generated modules over a PID and their applications to rational and Jordan canonical forms and the tensor products of modules. Throughout, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. In addition, the book contains many examples fully worked out and a variety of problems for practice and challenge. Solution to the odd-numbered problems are provided at the end of the book to encourage the student in problem solving. This new edition contains an introduction to categories and functors, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Noether-Lasker theorem. In addition, there are over 150 new problems and examples. This book provides a complete abstract algebra course, enabling instructors to select the topics for use in individual classes