约翰· B. 弗雷利（John B. Fraleigh）罗德岛大学数学与应用数学科学系荣休教授，一生致力于数学教育，出版过多本有影响力的图书，《抽象代数基础教程》是其代表作之一，这本书已经成为经典。
尼尔· 布兰德 （Neal Brand）北得克萨斯大学数学系荣休教授，曾被评为该校杰出教学教授。他曾担任美国数学协会得克萨斯分会理事，获得美国数学协会得克萨斯分会授予的杰出服务奖。
ADVANCED GROUP THEORY 145
16 Isomorphism Theorems 145
17 Sylow Theorems 149
18 Series ofGroups 157
19 Free Abelian Groups 166
20 Free Groups 172
21 Group Presentations 177
V
RINGS AND FIELDS 185
22 Rings and Fields 185
23 Integral Domains 194
24 Fermat’s and Euler’sTheorems 200
25 Encryption 205
VI
CONSTRUCTING RINGS AND FIELDS 211
26 TheFieldof Quotientsof anIntegral Domain 211
27 Rings of Polynomials 218
28 Factorization ofPolynomials over a Field 228
29 .AlgebraicCoding Theory 237
30 Homomorphisms andFactor Rings 243
31 Prime and MaximalIdeals 250
32 .Noncommutative Examples 258
VII
COMMUTATIVE ALGEBRA 267
33 Vector Spaces 267
34 UniqueFactorization Domains 275
35 Euclidean Domains 286
36 Number Theory 292
37 .Algebraic Geometry 297
38 .Gr¨obner Basesfor Ideals 303
VIII
EXTENSION FIELDS 311
39 IntroductiontoExtensionFields 311
40 AlgebraicExtensions 319
41 .GeometricConstructions 328
42 Finite Fields 335
Contents v
IX
GALOIS THEORY 341
43 Introductionto GaloisTheory 341 44 SplittingFields 349 45 SeparableExtensions 357 46 Galois Theory 364 47 Illustrations of Galois Theory 372 48 Cyclotomic Extensions 378 49 Insolvabilityof theQuintic 384
Appendix: Matrix Algebra 391 Bibliography 395 Notations 397 Answersto Odd-NumberedExercises Not Asking for De.nitions or Proofs 401
. Notrequiredfortheremainderofthetext.
. This sectionisa prerequisite forSections17 and36only.
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