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Abstract Algebra With Applications to Galois Theory Algebraic Geometry Representation Theory and Cryptography Third Edition by Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke ISBN 978-3111139517 3111139514

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Abstract Algebra With Applications to Galois Theory Algebraic Geometry Representation Theory and Cryptography Third Edition Gerhard Rosenberger Digital Instant Download

Author(s): Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke
ISBN(s): 9783111142524, 3111142523
Edition: 3
File Details: PDF, 8.41 MB
Year: 2024
Language: english
SKU: EB-58670864 Category: Tags: , ,

Abstract Algebra With Applications to Galois Theory Algebraic Geometry Representation Theory and Cryptography Third Edition by Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke- Ebook PDF Instant Download/Delivery:  978-3111139517, 3111139514
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Product details: 

ISBN 10:   3111139514

ISBN 13: 978-3111139517

Author: Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke

Abstract algebra is the study of algebraic structures like groups, rings and fields. This book provides an account of the theoretical foundations including applications to Galois Theory, Algebraic Geometry and Representation Theory. It implements the pedagogic approach to conveying algebra from the perspective of rings. The 3rd edition provides a revised and extended versions of the chapters on Algebraic Cryptography and Geometric Group Theory.

Table of contents: 

1. Groups, Rings, and Fields

  • 1.1 Abstract Algebra

  • 1.2 Rings

  • 1.3 Integral Domains and Fields

  • 1.4 Subrings and Ideals

  • 1.5 Factor Rings and Ring Homomorphisms

  • 1.6 Fields of Fractions

  • 1.7 Characteristic and Prime Rings

  • 1.8 Groups

  • 1.9 Exercises

2. Maximal and Prime Ideals

  • 2.1 Maximal and Prime Ideals

  • 2.2 Maximal and Prime Ideals of the Integers

  • 2.3 Prime Ideals and Integral Domains

  • 2.4 Maximal Ideals and Fields

  • 2.5 The Existence of Maximal Ideals

  • 2.6 Principal Ideals and Principal Ideal Domains

  • 2.7 Exercises

  • 2.8 Prime Elements and Unique Factorization Domains

  • 2.9 The Fundamental Theorem of Arithmetic

  • 2.10 Prime Elements, Units, and Irreducibles

3. Unique Factorization Domains

  • 3.1 Unique Factorization Domains

  • 3.2 Euclidean Domains

  • 3.3 Overview of Integral Domains

  • 3.4 Exercises

4. Polynomials and Polynomial Rings

  • 4.1 Degrees, Reducibility, and Roots

  • 4.2 Polynomial Rings over Fields

  • 4.3 Polynomial Rings over Integral Domains

  • 4.4 Polynomial Rings over Unique Factorization Domains

  • 4.5 Exercises

5. Field Extensions

  • 5.1 Extension Fields and Finite Extensions

  • 5.2 Finite and Algebraic Extensions

  • 5.3 Minimal Polynomials and Simple Extensions

  • 5.4 Algebraic Closures

  • 5.5 Algebraic and Transcendental Numbers

  • 5.6 Exercises

6. Constructible Numbers and Field Extensions

  • 6.1 Field Extensions and Compass and Straightedge Constructions

  • 6.2 Geometric Constructions

  • 6.3 Four Classical Construction Problems

    • 6.3.1 Squaring the Circle

    • 6.3.2 The Doubling of the Cube

    • 6.3.3 The Trisection of an Angle

    • 6.3.4 Construction of a Regular n-Gon

  • 6.4 Exercises

7. Kronecker’s Theorem and Algebraic Closures

  • 7.1 Kronecker’s Theorem

  • 7.2 Algebraic Closures and Algebraically Closed Fields

  • 7.3 The Fundamental Theorem of Algebra

  • 7.4 Splitting Fields

  • 7.5 Permutations and Symmetric Polynomials

  • 7.6 The Fundamental Theorem of Symmetric Polynomials

  • 7.7 Skew Field Extensions of C and the Frobenius Theorem

  • 7.8 Exercises

8. Splitting Fields and Normal Extensions

  • 8.1 Splitting Fields

  • 8.2 Normal Extensions

  • 8.3 Exercises

9. Examples of Groups

  • 9.1 Groups, Subgroups, and Examples

  • 9.2 Groups, Subgroups, and Isomorphisms

  • 9.3 Permutation Groups

  • 9.4 Cosets and Lagrange’s Theorem

  • 9.5 Generators and Cyclic Groups

  • 9.6 Exercises

10. Normal Subgroups, Factor Groups, and Direct Products

  • 10.1 Normal Subgroups and Factor Groups

  • 10.2 The Group Isomorphism Theorems

  • 10.3 Direct Products of Groups

11. Finite Abelian Groups

  • 11.1 Some Properties of Finite Groups

  • 11.2 Automorphisms of a Group

  • 11.3 Exercises

12. The Derived Series

  • 12.1 The Derived Series

  • 12.2 Exercises

13. The Sylow Theorems

  • 13.1 Group Actions

  • 13.2 Conjugacy Classes and the Class Equation

  • 13.3 Some Applications of the Sylow Theorems

  • 13.4 Exercises

14. Free Groups and Group Presentations

  • 14.1 Group Presentations and Combinatorial Group Theory

  • 14.2 Free Groups

  • 14.3 Group Presentations

  • 14.4 The Modular Group

  • 14.5 Exercises

15. Finite Galois Extensions

  • 15.1 Presentations of Subgroups

  • 15.2 Geometric Interpretation

  • 15.3 Presentations of Factor Groups

  • 15.4 Decision Problems

  • 15.5 Group Amalgams: Free Products and Direct Products

  • 15.6 Galois Theory and the Solvability of Polynomial Equations

  • 15.7 Automorphism Groups of Field Extensions

16. Separable Field Extensions

  • 16.1 Separability of Fields and Polynomials

  • 16.2 Perfect Fields

  • 16.3 Finite Fields

  • 16.4 Separable Extensions

  • 16.5 Separability and Galois Extensions

  • 16.6 The Primitive Element Theorem

  • 16.7 Exercises

17. Applications of Galois Theory

  • 17.1 Field Extensions by Radicals

  • 17.2 Cyclotomic Extensions

  • 17.3 Solvability and Galois Extensions

  • 17.4 The Insolvability of the Quintic Polynomial

  • 17.5 Constructibility of Regular n-Gons

  • 17.6 The Fundamental Theorem of Algebra

18. Modules over Principal Ideal Domains

  • 18.1 The Theory of Modules

  • 18.2 Modules over Rings

  • 18.3 Annihilators and Torsion

  • 18.4 Direct Products and Direct Sums of Modules

  • 18.5 Free Modules

  • 18.6 The Fundamental Theorem for Finitely Generated Modules

  • 18.7 Exercises

19. Finitely Generated Abelian Groups

  • 19.1 The Fundamental Theorem: p-Primary Components

  • 19.2 Exercises

20. Integral and Transcendental Extensions

  • 20.1 The Ring of Algebraic Integers

  • 20.2 Integral Ring Extensions

  • 20.3 Transcendental Field Extensions

  • 20.4 The Transcendence of e and π

  • 20.5 Exercises

21. The Hilbert Basis Theorem and the Nullstellensatz

  • 21.1 Algebraic Geometry

  • 21.2 Algebraic Varieties and Radicals

  • 21.3 The Hilbert Basis Theorem

  • 21.4 The Nullstellensatz

  • 21.5 Applications and Consequences of Hilbert’s Theorems

  • 21.6 Dimensions

  • 21.7 Exercises

22. Algebras and Group Representations

  • 22.1 Group Representations

  • 22.2 Representations and Modules

  • 22.3 Semisimple Algebras and Wedderburn’s Theorem

  • 22.4 Ordinary Representations, Characters, and Character Theory

  • 22.5 Burnside’s Theorem

23. Algebraic Cryptography

  • 23.1 Basic Algebraic Cryptography

  • 23.2 Cryptosystems Tied to Abelian Groups

  • 23.3 Cryptographic Protocols

  • 23.4 Non-Commutative Group-Based Cryptography

24. Group-Based Cryptosystems

  • 24.1 Group-Based Methods

  • 24.2 Initial Group Theoretic Cryptosystems – The Magnus Method

  • 24.2.1 The Wagner-Hungarian Method

  • 24.3 Free Group Cryptosystems

  • 24.4 Non-Abelian Digital Signature Procedure

  • 24.5 Password Authentication Using Combinatorial Group Theory

    • 24.5.1 General Outline of the Authentication Protocol

    • 24.5.2 Free Subgroup Method

    • 24.5.3 General Finitely Presented Group Method

  • 24.6 The Strong Generic Free Group Property

    • 24.6.1 Implementation of a Group Randomizer System Protocol

  • 24.7 Security Analysis of the Group Randomizer Protocols

  • 24.8 Ko-Lee and Anshel-Anshel-Goldfeld Protocols

    • 24.8.1 The Ko-Lee Protocol

    • 24.8.2 The Anshel-Anshel-Goldfeld Protocol

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Tags: Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke