Concepts in Abstract Algebra

Description:

The style and structure of CONCEPTS IN ABSTRACT ALGEBRA is designed to help students learn the core concepts and associated techniques in algebra deeply and well. Providing a fuller and richer account of material than time allows in a lecture, this text presents interesting examples of sufficient complexity so that students can see the concepts and results used in a nontrivial setting. Author Charles Lanski gives students the opportunity to practice by offering many exercises that require the use and synthesis of the techniques and results. Both readable and mathematically interesting, the text also helps students learn the art of constructing mathematical arguments. Overall, students discover how mathematics proceeds and how to use techniques that mathematicians actually employ. This book is included in the Brooks/Cole Series in Advanced Mathematics (Series Editor: Paul Sally, Jr.).

 

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.