Traditionally, mathematics has been separated into three main areas: algebra, anal-
ysis, and geometry. Of course, there is a great deal of overlap between these areas.
For example, topology, which is geometric in nature, owes its origins and problems
as much to analysis as to geometry. Furthermore, the basic techniques in studying
topology are predominantly algebraic. In general, algebraic methods and symbolism
pervadeall of mathematics,and it is essentialfor anyonelearninganyadvancedmath-
ematics to be familiar with the concepts and methods in abstract algebra.
Author(s): Celine Carstensen-Opitz, Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger
Edition: 2
Publisher: De Gruyter
Year: 2019
Language: English
Pages: 424
Cover......Page 1
Abstract Algebra
......Page 5
© 2019......Page 6
Preface......Page 7
Preface to the second edition......Page 9
Contents
......Page 11
1 Groups, rings and fields......Page 17
2 Maximal and prime ideals......Page 37
3 Prime elements and unique factorization domains......Page 45
4 Polynomials and polynomial rings......Page 69
5 Field extensions......Page 83
6 Field extensions and compass and straightedge
constructions......Page 97
7 Kronecker’s theorem and algebraic closures......Page 109
8 Splitting fields and normal extensions......Page 135
9 Groups, subgroups, and examples......Page 141
10 Normal subgroups, factor groups, and direct
products......Page 163
11 Symmetric and alternating groups......Page 185
12 Solvable groups......Page 195
13 Groups actions and the Sylow theorems......Page 205
14 Free groups and group presentations......Page 217
15 Finite Galois extensions......Page 243
16 Separable field extensions......Page 259
17 Applications of Galois theory......Page 273
18 The theory of modules......Page 291
19 Finitely generated Abelian groups......Page 309
20 Integral and transcendental extensions......Page 319
21 The Hilbert basis theorem and the nullstellensatz......Page 335
22 Algebras and group representations......Page 349
23 Algebraic cryptography......Page 381
Bibliography......Page 415
Index......Page 419
