Synopses & Reviews
This book provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. A repetition and examples first approach introduces learners to mathematical rigor and abstraction while teaching them the basic notions and results of modern algebra. Chapter topics include group theory, direct products and Abelian groups, rings and fields, geometric constructions, historical notes, symmetries, and coding theory. For future teachers of algebra and geometry at the high school level.
Synopsis
For a one-semester course covering groups and rings or a two-semester course in Abstract Algebra.This text provides thorough coverage of the main topics of abstract algebra while offering nearly 100 pages of applications. A repetition and examples first approach introduces students to mathematical rigor and abstraction while teaching them the basic notions and results of modern algebra.
Description
Includes bibliographical references (p. 539-540) and index.
Table of Contents
Preface.
Acknowledgments
0. Background.
Sets and Maps. Equivalence Relations and Partitions. Properties of Z. Complex Numbers. Matrices.
A. GROUP THEORY. 1. Groups.
Examples and Basic Concepts. Subgroups. Cyclic Groups. Permutations. 2. Group Homomorphisms.
Cosets and Lagrange's Theorem. Homomorphisms. Normal Subgroups. Quotient Groups. Automorphisms. 3. Direct Products and Abelian Groups.
Examples and Definitions. Computing Orders. Direct Sums. Fundamental Theorem of Finite Abelian Groups. 4. Group Actions.
Group Actions and Cayley's Theorem. Stabilizers and Orbits in a Group Action. Burnside's Theorem and Applications. Conjugacy Classes and the Class Equation. Conjugacy in Sn and Simplicity of A5. The Sylow Theorems. Applications of the Sylow Theorems. 5. Composition Series.
Isomorphism Theorems. The Jordan-Hölder Theorem. Solvable Groups.
B. RINGS AND FIELDS. 6. Rings.
Examples and Basic Concepts. Integral Domains. Fields. 7. Ring Homomorphisms.
Definitions and Basic Properties. Ideals. The Field of Quotients. 8. Rings of Polynomials.
Basic Concepts and Notation. The Division Algorithm in F[x]. More Applications of the Division Algorithm. Irreducible Polynomials. Cubic and Quartic Polynomials. Ideals in F[x]. Quotient Rings of F[x]. The Chinese Remainder Theorem for F[x]. 9. Euclidean Domains.
Division Algorithms and Euclidean Domains. Unique Factorization Domains. Gaussian Integers. 10. Field Theory.
Vector Spaces. Algebraic Extensions. Splitting Fields. Finite Fields. 11. Geometric Constructions.
Constructible Real Numbers. Classical Problems. Constructions with Marked Ruler and Compass. Cubics and Quartics Revisited. 12. Galois Theory.
Galois Groups. The Fundamental Theorem of Galois Theory. Galois Groups of Polynomials. Geometric Constructions Revisited. Radical Extensions. 13. Historical Notes.
From Ahmes the Scribe to Omar Khayyam. From Gerolamo Cardano to C. F. Gauss. From Evariste Galois to Emmy Noether.
C. SELECTED TOPICS. 14. Symmetries.
Linear Transformations. Isometries. Symmetry Groups. Platonic Solids. Subgroups of the Special Orthogonal Group. Further Reading. 15. Grobner Bases.
Lexicographic Order. A Division Algorithm. Dickson's Lemma. The Hilbert Basis Theorem. Gröbner Bases and the Division Algorithm. Further Reading. 16. Coding Theory.
Linear Binary Codes. Error Correction and Coset Decoding. Standard Generator Matrices. The Syndrome Method. Cyclic Codes. Further Reading. 17. Boolean Algebras.
Lattices. Boolean Algebras. Circuits. Further Reading. Answers and Hints to Selected Exercises.
Bibliography.
Index.