PROBLEMS IN ABSTRACT ALGEBRA.

• Chapter 1. Integers and Integers mod ??16
• Chapter 2. Groups22
2.1. Groups, subgroups, and cosets22
2.2. Group homomorphisms and factor groups34
2.3. Group actions41
2.4. Symmetric and alternating groups45
2.5. ??-groups50
2.6. Sylow subgroups52
2.7. Semidirect products of groups53
2.8. Free groups and groups by generators and relations62
2.9. Nilpotent, solvable, and simple groups67
2.10. Finite abelian groups75
• Chapter 3. Rings82
3.1. Rings, subrings, and ideals82
3.2. Factor rings and ring homomorphisms98
3.3. Polynomial rings and evaluation maps106
3.4. Integral domains, quotient fields109
3.5. Maximal ideals and prime ideals112
3.6. Divisibility and principal ideal domains116
3.7. Unique factorization domains124
• Chapter 4. Linear Algebra and Canonical Forms of Linear Transformations134
4.1. Vector spaces and linear dependence134
4.2. Linear transformations and matrices141
4.3. Dual space148
4.4. Determinants151
4.5. Eigenvalues and eigenvectors, triangulation and diagonalization159
4.6. Minimal polynomials of a linear transformation and primary decomposition164
4.7. ??-cyclic subspaces and ??-annihilators170
4.8. Projection maps173
4.9. Cyclic decomposition and rational and Jordan canonical forms176
4.10. The exponential of a matrix186
4.11. Symmetric and orthogonal matrices over \R189
4.12. Group theory problems using linear algebra196
• Chapter 5. Fields and Galois Theory200
5.1. Algebraic elements and algebraic field extensions201
5.2. Constructibility by compass and straightedge208
5.3. Transcendental extensions211
5.4. Criteria for irreducibility of polynomials214
5.5. Splitting fields, normal field extensions, and Galois groups217
5.6. Separability and repeated roots225
5.7. Finite fields232
5.8. Galois field extensions235
5.9. Cyclotomic polynomials and cyclotomic extensions243
5.10. Radical extensions, norms, and traces253
5.11. Solvability by radicals262

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.

Author
A. R. Wadsworth: University of California, San Diego, CA