Algebra

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 Zorn's 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 andEquivalence. - 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. Language: English