Abstract Algebra: A First Course

ISBN-13: 9781032289397 / Angielski / Twarda / 2022 / 557 str.

Stephen Lovett

Opis

This textbook intends to serve as a first course in abstract algebra. The selection of topics serves both of the common trends in such a course: a balanced introduction to groups, rings, and fields; or a course that primarily emphasizes group theory. This book offers a unique feature in the lists of projects at the end of each section.

Spis treści

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.

O autorze

Stephen Lovett is an associate professor of mathematics at Wheaton College. He earned a PhD from Northeastern University. He is a member of the Mathematical Association of America, American Mathematical Society, and Association of Christians in the Mathematical Sciences.. His research interests include commutative algebra, algebraic geometry, differential geometry, cryptography, and discrete dynamical systems. Dr. Lovett’s other books are: Differential Geometry of Curves and Surfaces, Third Edition with Thomas F. Banchoff, Differential Geometry of Manifolds, Second Edition, and Transition to Advanced Mathematics, with Danilo R. Diedrichs, all from CRC Press.