Problems in Abstract Algebra (Student Mathematical Library) (Student Mathematical Library, 82)

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......Page 1 Title page......Page 2 Contents......Page 4 Preface......Page 8 Introduction......Page 10 0.1. Notation......Page 12 0.2. Zorn’s Lemma......Page 14 Chapter 1. Integers and Integers mod ......Page 16 2.1. Groups, subgroups, and cosets......Page 22 2.2. Group homomorphisms and factor groups......Page 34 2.3. Group actions......Page 41 2.4. Symmetric and alternating groups......Page 45 2.5. -groups......Page 50 2.6. Sylow subgroups......Page 52 2.7. Semidirect products of groups......Page 53 2.8. Free groups and groups by generators and relations......Page 62 2.9. Nilpotent, solvable, and simple groups......Page 67 2.10. Finite abelian groups......Page 75 3.1. Rings, subrings, and ideals......Page 82 3.2. Factor rings and ring homomorphisms......Page 98 3.3. Polynomial rings and evaluation maps......Page 106 3.4. Integral domains, quotient fields......Page 109 3.5. Maximal ideals and prime ideals......Page 112 3.6. Divisibility and principal ideal domains......Page 116 3.7. Unique factorization domains......Page 124 4.1. Vector spaces and linear dependence......Page 134 4.2. Linear transformations and matrices......Page 141 4.3. Dual space......Page 148 4.4. Determinants......Page 151 4.5. Eigenvalues and eigenvectors, triangulation and diagonalization......Page 159 4.6. Minimal polynomials of a linear transformation and primary decomposition......Page 164 4.7. -cyclic subspaces and -annihilators......Page 170 4.8. Projection maps......Page 173 4.9. Cyclic decomposition and rational and Jordan canonical forms......Page 176 4.10. The exponential of a matrix......Page 186 4.11. Symmetric and orthogonal matrices over \R......Page 189 4.12. Group theory problems using linear algebra......Page 196 Chapter 5. Fields and Galois Theory......Page 200 5.1. Algebraic elements and algebraic field extensions......Page 201 5.2. Constructibility by compass and straightedge......Page 208 5.3. Transcendental extensions......Page 211 5.4. Criteria for irreducibility of polynomials......Page 214 5.5. Splitting fields, normal field extensions, and Galois groups......Page 217 5.6. Separability and repeated roots......Page 225 5.7. Finite fields......Page 232 5.8. Galois field extensions......Page 235 5.9. Cyclotomic polynomials and cyclotomic extensions......Page 243 5.10. Radical extensions, norms, and traces......Page 253 5.11. Solvability by radicals......Page 262 Suggestions for Further Reading......Page 266 Bibliography......Page 268 Index of Notation......Page 270 Subject and Terminology Index......Page 276 Back Cover......Page 290 Cover -- Title page -- Contents -- Preface -- Introduction -- 0.1. Notation -- 0.2. Zorn's Lemma -- Chapter 1. Integers and Integers mod -- 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.-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.-cyclic subspaces and -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 Presents problems in abstract algebra for strong undergraduates or beginning graduate students. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations); rings (including basic ideal theory and factorization in integral domains); linear algebra (emphasizing linear transformations); and fields (including Galois theory).