دانلود کتاب Abstract Algebra: With Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography (De Gruyter Textbook)

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cover
Preface
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 of the Integers
2.2 Prime Ideals and Integral Domains
2.3 Maximal Ideals and Fields
2.4 The Existence of Maximal Ideals
2.5 Principal Ideals and Principal Ideal Domains
2.6 Exercises
3 Prime Elements and Unique Factorization Domains
3.1 The Fundamental Theorem of Arithmetic
3.2 Prime Elements, Units and Irreducibles
3.3 Unique Factorization Domains
3.4 Principal Ideal Domains and Unique Factorization
3.5 Euclidean Domains
3.6 Overview of Integral Domains
3.7 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 Field Extensions and Compass and Straightedge Constructions
6.1 Geometric Constructions
6.2 Constructible Numbers and Field Extensions
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
6.8 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
7.3.1 Splitting Fields
7.3.2 Permutations and Symmetric Polynomials
7.4 The Fundamental Theorem of Symmetric Polynomials
7.5 Skew Field Extensions of ℂ and the Frobenius Theorem
7.6 Exercises
8 Splitting Fields and Normal Extensions
8.1 Splitting Fields
8.2 Normal Extensions
8.3 Exercises
9 Groups, Subgroups and Examples
9.1 Groups, Subgroups and Isomorphisms
9.2 Examples of Groups
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
10.4 Finite Abelian Groups
10.5 Some Properties of Finite Groups
10.6 Automorphisms of a Group
10.7 Exercises
11 Symmetric and Alternating Groups
11.1 Symmetric Groups and Cycle Decomposition
11.2 Parity and the Alternating Groups
11.3 The Conjugation in Sn
11.4 The Simplicity of An
11.5 Exercises
12 Solvable Groups
12.1 Solvability and Solvable Groups
12.2 The Derived Series
12.3 Composition Series and the Jordan–Hölder Theorem
12.4 Exercises
13 Group Actions and the Sylow Theorems
13.1 Group Actions
13.2 Conjugacy Classes and the Class Equation
13.3 The Sylow Theorems
13.4 Some Applications of the Sylow Theorems
13.5 Exercises
14 Free Groups and Group Presentations
14.1 Group Presentations and Combinatorial Group Theory
14.2 Free Groups
14.3 Group Presentations
14.3.1 The Modular Group
14.4 Presentations of Subgroups
14.5 Geometric Interpretation
14.6 Presentations of Factor Groups
14.7 Decision Problems
14.8 Group Amalgams: Free Products and Direct Products
14.9 Exercises
15 Finite Galois Extensions
15.1 Galois Theory and the Solvability of Polynomial Equations
15.2 Automorphism Groups of Field Extensions
15.3 Finite Galois Extensions
15.4 The Fundamental Theorem of Galois Theory
15.5 Exercises
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
17.7 Exercises
18 The Theory of Modules
18.1 Modules over Rings
18.2 Annihilators and Torsion
18.3 Direct Products and Direct Sums of Modules
18.4 Free Modules
18.5 Modules over Principal Ideal Domains
18.6 The Fundamental Theorem for Finitely Generated Modules
18.7 Exercises
19 Finitely Generated Abelian Groups
19.1 Finite Abelian Groups
19.2 The Fundamental Theorem: p-Primary Components
19.3 The Fundamental Theorem: Elementary Divisors
19.4 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
22.6 Exercises
23 Algebraic Cryptography
23.1 Basic Algebraic Cryptography
23.1.1 Cryptosystems Tied to Abelian Groups
23.1.2 Cryptographic Protocols
24 Non-Commutative Group Based Cryptography
24.1 Group Based Methods
24.2 Initial Group Theoretic Cryptosystems—The Magnus Method
24.2.1 The Wagner–Magyarik 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 Security Analysis of the Group Randomizer Protocols
24.6.2 Implementation of a Group Randomizer System Protocol
24.7 A Secret Sharing Scheme Using Combinatorial Group Theory
24.8 Ko–Lee and Anshel–Anshel–Goldfeld Protocols
24.8.1 The Ko–Lee Protocol
24.8.2 The Anshel–Anshel–Goldfeld Protocol
Bibliography
Index