tailieunhanh - Ebook A first course in abstract algebra (7th edition): Part 1

(BQ) Considered a classic by many "A first course in abstract algebra" focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. | 7 Contents Instructor s Preface vii Student s Preface xi Dependence Chart xiii 0 Sets and Relations 1 I Groups and Subgroups 11 1 Introduction and Examples 11 2 Binary Operations 20 3 Isomorphic Binary Structures 28 4 Groups 36 - 5 Subgroups 49 i 6 Cyclic Groups 59 7 Generating Sets and Cayley Digraphs 68 II Permutations Cosets and Direct Products 75 8 Groups of Permutations 75 9 Orbits Cycles and the Alternating Groups 87 10 Cosets and the Theorem of Lagrange 96 11 Direct Products and Finitely Generated Abelian Groups 104 12 Plane Isometries 114 iỉi iv Contents III Homomorphisms and Factor Groups 125 13 Homomorphisms 125 14 Factor Groups 135 15 Factor-Group Computations and Simple Groups 144 Í16 Group Action on a Set 154 tl7 Applications of G-Sets to Counting 161 IV Rings and Fields 167 18 Rings and Fields 167 19 Integral Domains 177 20 Fermat s and Euler s Theorems 184 21 The Field of Quotients of an Integral Domain 190 22 Rings of Polynomials 198 23 Factorization of Polynomials over a Field 209 T24 Noncommutative Examples 220 t25 Ordered Rings and Fields 227 Ideals and Factor Rings 237 26 Homomorphisms and Factor Rings 237 27 Prime and Maximal Ideals 245 t28 Grobner Bases for Ideals 254 VI Extension Fields 265 29 Introduction to Extension Fields 265 30 Vector Spaces 274 31 Algebraic Extensions 283 32 Geometric Constructions 293 33 Finite Fields 300 VII Advanced Group Theory 307 34 Isomorphism Theorems 307 35 Series of Groups 311 36 Sylow Theorems 321 37 Applications of the Sylow Theory 327 Contents V 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Free Abelian Groups 333 Free Groups 341 Group Presentations 346 VIII Groups in Topology 355 Simplicial Complexes and Homology Groups 355 Computations of Homology Groups 363 More Homology Computations and Applications 371 Homological Algebra 379 Factorization Unique Factorization Domains 389 Euclidean Domains 401 Gaussian Integers and Multiplicative Norms 407 389 X Automorphisms and Galois Theory 415 .