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

ISBN-13: 9783110250084 / Angielski / Twarda / 2011 / 366 str.

Celine Carstensen; Gerhard Rosenberger; Benjamin Fine

Opis

A new approach to conveying abstract algebra, the area that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, that is essential to various scientific disciplines such as particle physics and cryptology. It provides a well written account of the theoretical foundations; also contains topics that cannot be found elsewhere, and also offers a chapter on cryptography. End of chapter problems help readers with accessing the subjects. This work is co-published with the Heldermann Verlag, and within Heldermann's Sigma Series in Mathematics."

Spis treści



1 Groups, Rings and Fields

2 Maximal and Prime Ideals

3 Prime Elements and Unique Factorization Domains

4 Polynomials and Polynomial Rings

5 Field Extensions

6 Field Extensions and Compass and Straightedge Constructions

7 Kronecker's Theorem and Algebraic Closures

8 Splitting Fields and Normal Extensions

9 Groups, Subgroups and Examples

10 Normal Subgroups, Factor Groups and Direct Products

11 Symmetric and Alternating Groups

12 Solvable Groups

13 Groups Actions and the Sylow Theorems

14 Free Groups and Group Presentations

15 Finite Galois Extensions

16 Separable Field Extensions

17 Applications of Galois Theory

18 The Theory of Modules

19 Finitely Generated Abelian Groups

20 Integral and Transcendental Extensions

21 The Hilbert Basis Theorem and the Nullstellensatz

22 Algebraic Cryptography