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

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