Solutions Manual for Herstein's Topics in Algebra (代數特論問題詳解)

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Contents 1. Preliminary Notions 1.1 Set Theory p.8 p.9 1.2 Mappings p.16 p.17 1.3 The Integers p.23 p.24 2. Group Theory 2.3 Some Preliminary Lemmas p.35 p.36 p.37 2.5 A Counting Principle p.46 p.47 p.48 p.49 2.6 Normal Subgroups and Quotient Groups p.53 p.54 2.7 Homomorphisms p.64 p.65 p.66 2.8 Automorphisms p.70 p.71 2.9 Cayley's Theorem p.74 p.75 2.10 Permutation Groups p.80 p.81 p.82 2.11 Another Counting Principle p.90 p.91 2.12 Sylow's Theorem p.101 p.102 p.103 2.13 Direct Products p.108 p.109 2.14 Finite Abelian Groups p.115 p.116 p.117 p.118 p.119 A CLASSIFICATION OF n-ABELIAN GROUPS -- J. L. Alperin MATHEMATICAL NOTES THE AUTOMORPHISMS OF THE SYMMETRIC GROUP -- Irving K. Segal 3. Ring Theory 3.2 Some Special C lasses of Rings p.130 3.4 Idea ls and Quotient Rings p.135 p.136 p.137 3.5 More Ideals and Quotient Rings p.139 p.140 3.6 Field of Quotients of lntegeral Domain p.142 p.143 3.7 Euclidean Rings p.149 3.8 A Particular Euclidean Ring p.152 3.9 Polynomial Rings p.158 p.159 3.10 Polynomials over Rational Field p.161 3.11 Polynomial Rings over Commutative Rings p.166 Supplementary Problems p.167 p.168 THE EUCLIDEAN ALGORlTHM -- TH. Motzkin 4. Vector Spaces and Modules 4.1 Elementary Basic Concepts p.175 p.176 p.177 4.2 Linear Independence and Bases p.183 p.184 4.3 Dual Spaces p.190 p.191 4.4. Inner Product Spaces p.199 p.200 4.5 modules p.205 p.206 5. Fields 5.1 Extension Fields p.214 p.215 p.216 5.2 Transcendence of e p.219 5.3 Roots of Polynomials p.227 p.228 5.4 Construction with Straighedge and Compass p.231 p.232 5.5 More About Roots p.236 p.237 5.6 Elements of Galois Theory p.249 p.250 5.7 Solvability by Radicals p.256 5.8 Galois Groups over the Rationals p.259 6. Linear Transformations 6.1 Algebra of Linear Transformations p.267 p.268 p.269 6.2 Characteristic Roots p.272 p.273 6.3 Matrices p.281 p.282 p.283 p.284 p.285 6.4 Canonical Forms: Triangular Form p.290 p.291 p.292 6.6 Canonical Forms : Decomposition of V: Jordan Form p.303 p.304 p.305 6.7 Canonical Forms: Rational Canonical Form p.312 p.313 6.8 Trace and Transpose p.319 p.320 p.321 p.322 6.9 Determinants p.334 p.335 p.336 6.10 Hermitian, Unitary, and Normal Transformations p.349 p.350 6.11 Real Quadratic Forms p.354 7. Selected Topics 7.1 Finite Fields p.360 7.2 Wedderburn's Theorem on Finite Division Rings p.367 p.368 7.3 A Theorem of Frobenius p.371 7.4 Integral Quaternions and the Four-Square Theorem p.377