AlgebraAlgebra fulfills a definite need to provide a self-contained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially self-contained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth. |
Contents
Groups | 23 |
1 Semigroups Monoids and Groups | 24 |
2 Homomorphisms and Subgroups | 30 |
3 Cyclic Groups | 35 |
4 Cosets and Counting | 37 |
5 Normality Quotient Groups and Homomorphisms | 41 |
6 Symmetric Alternating and Dihedral Groups | 46 |
Products Coproducts and Free Objects | 52 |
4 The Galois Group of a Polynomial | 269 |
5 Finite Fields | 278 |
6 Separability | 282 |
7 Cyclic Extensions | 289 |
8 Cyclotomic Extensions | 297 |
9 Radical Extensions | 302 |
The Structure of Fields | 311 |
2 Linear Disjointness and Separability | 318 |
8 Direct Products and Direct Sums | 59 |
9 Free Groups Free Products Generators Relations | 64 |
The Structure of Groups | 70 |
2 Finitely Generated Abelian Groups | 76 |
3 The KrullSchmidt Theorem | 83 |
4 The Action of a Group on a Set | 88 |
5 The Sylow Theorems | 92 |
6 Classification of Finite Groups | 96 |
7 Nilpotent and Solvable Groups | 100 |
8 Normal and Subnormal Series | 107 |
Rings | 114 |
1 Rings and Homomorphisms | 115 |
2 Ideals | 122 |
3 Factorization in Commutative Rings | 135 |
4 Rings of Quotients and Localization | 142 |
5 Ring of Polynomials and Formal Power Series | 149 |
6 Factorization in Polynomial Rings | 157 |
Modules | 168 |
1 Modules Homomorphisms and Exact Sequences | 169 |
2 Free Modules and Vector Spaces | 180 |
3 Projective and Injective Modules | 190 |
4 Hom and Duality | 199 |
5 Tensor Products | 207 |
6 Modules Over a Principal Ideal Domain | 218 |
7 Algebras | 226 |
Fields and Galois Theory | 230 |
1 Field Extensions | 231 |
2 The Fundamental Theorem | 243 |
3 Splitting Fields Algebraic Closure and Normality | 257 |
Linear Algebra | 327 |
1 Matrices and Maps | 328 |
2 Rank and Equivalence | 335 |
3 Determinants | 348 |
4 Decomposition of a Single Linear Transformation and Similarity | 355 |
5 The Characteristic Polynomial Eigenvectors and Eigenvalues | 366 |
Commutative Rings and Modules | 371 |
1 Chain Conditions | 372 |
2 Prime and Primary Ideals | 377 |
3 Primary Decomposition | 383 |
4 Noetherian Rings and Modules | 387 |
5 Ring Extensions | 394 |
6 Dedekind Domains | 400 |
7 The Hilbert Nullstellensatz | 409 |
The Structure of Rings | 414 |
1 Simple and Primitive Rings | 415 |
2 The Jacobson Radical | 424 |
3 Semisimple Rings | 434 |
4 The Prime Radical Prime and Semiprime Rings | 444 |
5 Algebras | 450 |
6 Division Algebras | 456 |
Categories | 464 |
1 Functors and Natural Transformations | 465 |
2 Adjoint Functors | 476 |
3 Morphisms | 480 |
List of Symbols | 485 |
| 489 | |
| 493 | |
Common terms and phrases
a₁ abelian group algebraically closed automorphism b₁ basis chain condition char commutative ring Consequently contains Corollary coset cyclic defined Definition denoted direct sum disjoint division ring divisors element endomorphism epimorphism equivalent EXAMPLE Exercise exists extension field factors finite dimensional free module function functor G₁ Galois group given group G hence Hint implies infinite integral domain intermediate field invertible irreducible isomorphism K-algebra left Artinian left ideal left R-module Lemma Let F Let G linear linearly independent matrix monic monomorphism morphism multiplicative nilpotent Noetherian nonempty nonzero normal subgroup P₁ phism polynomial positive integer prime ideal primitive principal ideal domain Proposition prove quotient R-module R-module homomorphism r₁ radical resp ring with identity root Section semisimple SKETCH OF PROOF solvable splitting field subfield subgroup of G submodule subring subset Sylow Theorem 1.6 u₁ unique vector space Verify whence zero