Abstract Algebra : A First Course (Paperback, 2 ed)

1. Groups. 1.1. Symmetries of a Regular Polygon. 1.2. Introduction to Groups. 1.3. Properties of Group Elements. 1.4. Concept of a Classification Theorem. 1.5. Symmetric Groups. 1.6. Subgroups. 1.7. Abstract Subgroups. 1.8. Lattice of Subgroups. 1.9. Group Homomorphisms. 1.10. Group Presentations. 1.11. Groups in Geometry. 1.12. Diffie-Hellman Public Key. 1.13. Semigroups and Monoids. 1.14. Projects. 2. Quotient Groups. 2.1. Cosets and Lagrange’s Theorem. 2.2. Conjugacy and Normal Subgroups. 2.3. Quotient Groups. 2.4. Isomorphism Theorems. 2.5. Fundamental Theorem of Finitely Generated Abelian Groups. 2.6. Projects. 3. Rings. 3.1. Introduction to Rings. 3.2. Rings Generated by Elements. 3.3. Matrix Rings. 3.4. Ring Homomorphisms. 3.5. Ideals. 3.6. Operations on Ideals. 3.7. Quotient Rings. 3.8. Maximal Ideals and Prime Ideals. 3.9. Projects. 4. Divisibility in Integral Domains. 4.1. Divisibility in Commutative Rings. 4.2. Rings of Fractions. 4.3. Euclidean Domains. 4.4. Unique Factorization Domains. 4.5. Factorization of Polynomials. 4.6. RSA Cryptography. 4.7. Algebraic Integers. 4.8. Projects. 5. Field Extensions. 5.1. Introduction to Field Extensions. 5.2. Algebraic and Transcendental Elements. 5.3. Algebraic Extensions. 5.4. Solving Cubic and Quartic Equations. 5.5. Constructible Numbers. 5.6. Cyclotomic Extensions. 5.7. Splitting Fields and Algebraic Closure. 5.8. Finite Fields. 5.9. Projects. 6. Topics in Group Theory. 6.1. Introduction to Group Actions. 6.2. Orbits and Stabilizers. 6.3. Transitive Group Actions. 6.4. Groups Acting on Themselves. 6.5. Sylow’s Theorem. 6.6. Semidirect Product. 6.7. Classification Theorems. A. Appendix. Bibliography. Index.