Concepts in abstract algebra

Synopsis

0. REVIEW. Sets. Index Sets and Partitions. Induction. Well Ordering and Induction. Functions. Bijections and Inverses. Cardinality and Infinite Sets. 1. PRELIMINARIES. Remainders. Divisibility. Relative Primeness. Prime Factorization. Relations. Equivalence Relations. Congruence Modulo n. The Ring of Integers Mod n. Localization. 2. GROUPS. Basic Notions and Examples. Uniqueness Properties. Groups of Symmetries. Orders of Elements. Subgroups. Special Subgroups. 3. SPECIAL GROUPS. Cyclic Groups. The Groups Un. The Symmetric Groups Sn. The Dihedral Groups Dn. Direct Sums. 4. SUBGROUPS. Cosets. Lagrange's Theorem and Consequences. Products of Subgroups. Products in Abelian Groups. Cauchy's Theorem and Cyclic Groups. The Groups Up are Cyclic. Carmichael Numbers. Encryption and Codes. 5. NORMAL SUBGROUPS AND QUOTIENTS. Normal Subgroups. Quotient Groups. Some Results Using Quotient Groups. Simple Groups. 6. MORPHISMS. Homomorphisms. Basic Results. The First Isomorphism Theorem. Applications. Automorphisms. 7. STRUCTURE THEOREMS. The Correspondence Theorem. Two Isomorphism Theorems. Direct Sum Decompositions. Groups of Small Order. Fundamental Theorem of Finite Abelian Groups. 8. CONJUGATION. Conjugates. Conjugates and Centralizers in Sn. p-Groups. Sylow Subgroups. 9. GROUP ACTIONS. Group Actions. Counting Orbits. Sylow's Theorems. Applications of Sylow's Theorems. 10. RINGS. Definitions and Examples. Subrings. Polynomial and Related Rings. Zero Divisors and Domains. Indeterminates as Functions. 11. IDEALS, QUOTIENTS, AND HOMOMORPHISMS. Ideals. Quotient Rings. Homomorphisms of Rings. Isomorphism Theorems. The Correspondence Theorem. Chain Conditions. 12. FACTORIZATION IN INTEGRAL DOMAINS. Primes and Irreducibles. PIDs. UFDs. Euclidean Domains. 13. COMMUTATIVE RINGS. Maximal and Prime Ideals. Localization Revisited. Noetherian Rings. Integrality. Algebraic Geometry. Zorn's Lemma and Cardinality. 14. FIELDS. Vector Spaces. Subfields. Geometric Constructions. Splitting Fields. Algebraic Closures. Transcendental Extensions. 15. GALOIS THEORY. The Galois Correspondence. The Fundamental Theorem. Applications. Cyclotomic Extensions. Solvable Groups. Radical Extensions. Hints to Selected Odd Numbered Problems. Index.