Abstract Algebra: Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

Description

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. 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