Author | Hungerford, Thomas W. author |
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Title | Algebra [electronic resource] / by Thomas W. Hungerford |
Imprint | New York, NY : Springer New York, 1974 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6101-8 |
Descript | XXIV, 504 p. online resource |
Introduction: Prerequisites and Preliminaries -- 1. Logic -- 2. Sets and Classes -- 3. Functions -- 4. Relations and Partitions -- 5. Products -- 6. The Integers -- 7. The Axiom of Choice, Order and Zornโs Lemma -- 8. Cardinal Numbers -- I: Groups -- 1. Semigroups, Monoids and Groups -- 2. Homomorphisms and Subgroups -- 3. Cyclic Groups -- 4. Cosets and Counting -- 5. Normality, Quotient Groups, and Homomorphisms -- 6. Symmetric, Alternating, and Dihedral Groups -- 7. Categories: Products, Coproducts, and Free Objects -- 8. Direct Products and Direct Sums -- 9. Free Groups, Free Products, Generators & Relations -- II: The Structure of Groups -- 1. Free Abelian Groups -- 2. Finitely Generated Abelian Groups -- 3. The Kruli-Schmidt Theorem -- 4. The Action of a Group on a Set -- 5. The Sylow Theorems -- 6. Classification of Finite Groups -- 7. Nilpotent and Solvable Groups -- 8. Normal and Subnormal Series -- III: Rings -- 1. Rings and Homomorphisms -- 2. Ideals -- 3. Factorization in Commutative Rings -- 4. Rings of Quotients and Localization -- 5. Rings of Polynomials and Formal Power Series -- 6. Factorization in Polynomial Rings -- IV: Modules -- 1. Modules, Homomorphisms and Exact Sequences -- 2. Free Modules and Vector Spaces -- 3. Projective and Injective Modules -- 4. Horn and Duality -- 5. Tensor Products -- 6. Modules over a Principal Ideal Domain -- 7. Algebras -- V: Fields and Galois Theory -- 1. Field Extensions -- 2. The Fundamental Theorem -- 3. Splitting Fields, Algebraic Closure and Normality -- 4. The Galois Group of a Polynomial -- 5. Finite Fields -- 6. Separability -- 7. Cyclic Extensions -- 8. Cyclotomic Extensions -- 9. Radical Extensions -- VI: The Structure of Fields -- 1. Transcendence Bases -- 2. Linear Disjointness and Separability -- VII: Linear Algebra -- 1. Matrices and Maps -- 2. Rank and Equivalence -- 3. Determinants -- 4. Decomposition of a Single Linear Transformation and Similarity -- 5. The Characteristic Polynomial, Eigenvectors and Eigenvalues -- VIII: Commutative Rings and Modules -- 1. Chain Conditions -- 2. Prime and Primary Ideals -- 3. Primary Decomposition -- 4. Noetherian Rings and Modules -- 5. Ring Extensions -- 6. Dedekind Domains -- 7. The Hilbert Nullstellensatz -- IX: The Structure of Rings -- 1. Simple and Primitive Rings -- 2. The Jacobson Radical -- 3. Semisimple Rings -- 4. The Prime Radical; Prime and Semiprime Rings -- 5. Algebras -- 6. Division Algebras -- X: Categories -- 1. Functors and Natural Transformations -- 2. Adjoint Functors -- 3. Morphisms -- List of Symbols