Basic Abstract Algebra

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. 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 402 4. Uniform modules, primary modules, and Noether?asker theorem ......Page 408 1. Preliminaries ......Page 412 2. Row module, column module, and rank ......Page 413 3. Smith normal form......Page 414 1. Decomposition theorem ......Page 422 2. Uniqueness of the decomposition ......Page 424 3. Application to finitely generated abelian groups ......Page 428 4. Rational canonical form ......Page 429 5. Generalized Jordan form over any field ......Page 437 1. Categories and functors ......Page 445 2. Tensor products ......Page 447 3. Module structure of tensor product ......Page 450 4. Tensor product of homomorphisms ......Page 452 5. Tensor product of algebras......Page 455 Solutions to odd-numbered problems ......Page 457 Selected bibliography......Page 495 Index ......Page 496 Back Cover......Page 507 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 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. This book provides a complete abstract algebra course, enabling instructors to select the topics for use in individual classes