Problems in Abstract Algebra (Student Mathematical Library)

Description

This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included. Cover Title page Contents Preface Introduction 0.1. Notation 0.2. Zorn’s Lemma Chapter 1. Integers and Integers mod 𝑛 Chapter 2. Groups 2.1. Groups, subgroups, and cosets 2.2. Group homomorphisms and factor groups 2.3. Group actions 2.4. Symmetric and alternating groups 2.5. 𝑝-groups 2.6. Sylow subgroups 2.7. Semidirect products of groups 2.8. Free groups and groups by generators and relations 2.9. Nilpotent, solvable, and simple groups 2.10. Finite abelian groups Chapter 3. Rings 3.1. Rings, subrings, and ideals 3.2. Factor rings and ring homomorphisms 3.3. Polynomial rings and evaluation maps 3.4. Integral domains, quotient fields 3.5. Maximal ideals and prime ideals 3.6. Divisibility and principal ideal domains 3.7. Unique factorization domains Chapter 4. Linear Algebra and Canonical Forms of Linear Transformations 4.1. Vector spaces and linear dependence 4.2. Linear transformations and matrices 4.3. Dual space 4.4. Determinants 4.5. Eigenvalues and eigenvectors, triangulation and diagonalization 4.6. Minimal polynomials of a linear transformation and primary decomposition 4.7. 𝑇-cyclic subspaces and 𝑇-annihilators 4.8. Projection maps 4.9. Cyclic decomposition and rational and Jordan canonical forms 4.10. The exponential of a matrix 4.11. Symmetric and orthogonal matrices over \R 4.12. Group theory problems using linear algebra Chapter 5. Fields and Galois Theory 5.1. Algebraic elements and algebraic field extensions 5.2. Constructibility by compass and straightedge 5.3. Transcendental extensions 5.4. Criteria for irreducibility of polynomials 5.5. Splitting fields, normal field extensions, and Galois groups 5.6. Separability and repeated roots 5.7. Finite fields 5.8. Galois field extensions 5.9. Cyclotomic polynomials and cyclotomic extensions 5.10. Radical extensions, norms, and traces 5.11. Solvability by radicals Suggestions for Further Reading Bibliography Index of Notation Subject and Terminology Index Back Cover