Abstract Algebra

Title Preface Contents 0. Preliminaries 0.1. Set theoretic notations 0.2. Correspondences, maps, relations 0.3. Cartesian product and universal mapping properties 0.4. Cardinal numbers 0.5. Zorn's lemma, axiom of choice, well-ordering axiom References I. Natural Numbers, Integers, and Rational Numbers I.1. Peano's axioms I.2. Addition in P I.3. Ordering in P: A second definition of finiteness I.4. Multiplication in P I.5. Construction of the integers Z I.6. Divisibility theory in Z I.7. Construction of the rational numbers Q References II. Groups, Rings, Integral Domains, Fields II.1. Multiplicative systems: Groups II.2. Homomorphisms II.3. Rings, integral domains, and fields II.4. Polynomial rings References III. Elementary Theory of Groups III.1. Basic concepts III.2. Homomorphisms of groups Excursion I III.3. Transformation groups: Sylow's theorem III.4. The finite symmetric groups Sn Excursion II III.5. Direct product of groups: Fundamental Theorem of Finite Abelian Groups Excursion III References IV. Elementary Theory of Rings IV.1. Basic concepts IV.2. Divisibility theory in integral domains IV.3. Fields of rational functions: Partial fraction decomposition IV.4. Modules and their endomorphism rings. Matrices IV.5. Rings of functions References V. Modules and Associated Algebras over Commutative Rings V.1. Tensor product of modules over commutative rings V.2. Free modules over a PID Excursion IV V.3. Modules of finite type over a PID Excursion V V.4. Tensor algebras, exterior algebras, and determinants V.5. Derivations, traces, and characteristic polynomials Excursion VI V.6. Dual modules References VI. Vector Spaces VI.1. Basic concepts VI.2. Systems of linear equations VI.3. Decomposition of a vector space with respect to a linear endomorphism VI.4. Canonical forms of matrices: Characteristic values: Characteristic vectors Excursion VII References VII. Elementary Theory of Fields VII.1. Basic concepts VII.2. Algebraic extension fields: Splitting fields VII.3. Algebraically closed fields: Algebraic closure VII.4. Algebraic independence: Purely transcendental extensions: Transcendence base Excursion VIII VII.5. Separable and inseparable algebraic field extensions VII.6. Finite fields: Primitive element theorem References VIII. Galois Theory VIII.1. Basic concepts VIII.2. Fundamental theorems Excursion IX Excursion X VIII.3. Solvability of polynomial equations by radicals VIII.4. Cyclotomic polynomials over Q: Kummer extensions References IX. Real and Complex Numbers IX.1. Construction of real and complex numbers IX.2. Fundamental Theorem of Algebra Excursion XI Excursion XII References Index Symbols and notations