- #1
- 22,183
- 3,322
- Author: Charles Pinter
- Title: A book of Abstract Algebra
- Amazon link https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20
- Prerequisities: High-school algebra
- Level: Undergrad
Table of Contents:
Code:
[LIST]
[*] Preface
[*] Why Abstract Algebra?
[LIST]
[*] History of Algebra
[*] New Algebras
[*] Algebraic Structures
[*] Axioms and Axiomatic Algebra
[*] Abstraction in Algebra
[/LIST]
[*] Operations
[LIST]
[*] Operations on a Set
[*] Properties of Operations
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[*] The Definition of Groups
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[*] Groups
[*] Examples of Infinite and Finite Groups
[*] Examples of Abelian and Nonabelian Groups
[*] Group Tables
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[*] Elementary Properties of Groups
[LIST]
[*] Uniqueness of Identity and Inverses
[*] Properties of Inverses
[*] Direct Product of Groups
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[*] Subgroups
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[*] Definition of Subgroups
[*] Generators and Defining Relations
[*] Cayley Diagrams
[*] Center of a Group
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[*] Functions
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[*] Injective, Surjective, Bijective Function
[*] Composite and Inverse of Functions
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[*] Groups of Permutations
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[*] Symmetric Groups
[*] Dihedral Groups
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[*] Permutations of a Finite Set
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[*] Decomposition of Permutations into Cycles
[*] Transpositions
[*] Even and Odd Permutations
[*] Alternating Groups
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[*] Isomorphism
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[*] The Concept of Isomorphism in Mathematics
[*] Isomorphic and Nonisomorphic Groups
[*] Cayley's Theorem
[*] Group Automorphisms
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[*] Order of Groups Elements
[LIST]
[*] Powers/Multiples of Group Elements
[*] Laws of Exponents
[*] Properties of the Order of Group Elements
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[*] Cyclic Groups
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[*] Finite and Infinite Cyclic Groups
[*] Isomorphism of Cyclic Groups
[*] Subgroups of Cyclic Groups
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[*] Partitions and Equivalence Relations
[*] Counting Cosets
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[*] Lagrange's Theorem and Elementary Consequences
[*] Number of Conjugate Elements
[*] Group Acting on a Set
[*] Survey of Groups of Order [itex]\leq 10[/itex]
[/LIST]
[*] Homomorphisms
[LIST]
[*] Elementary Properties of Homomorphism
[*] Normal Subgroups
[*] Kernel and Range
[*] Inner Direct Products
[*] Conjugate Subgroups
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[*] Quotient Groups
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[*] Quotient Group Construction
[*] Examples and Applications
[*] The Class Equation
[*] Induction on the Order of a Group
[/LIST]
[*] The Fundamental Homomorphism Theorem
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[*] Fundamental Homomorphism Theorem and Some Consequences
[*] The Isomorphism Theorems
[*] The Correspondence Theorem
[*] Cauchy's Theorem
[*] Sylow Subgroups
[*] Sylow's Theorem
[*] Decomposition Theorem for Finite Abelian Groups
[/LIST]
[*] Rings: Definitions and elementary Properties
[LIST]
[*] Commutative Rings
[*] Unity
[*] Invertibles and Zero-Divisors
[*] Integral Domain
[*] Field
[/LIST]
[*] Ideal and Homomorphisms
[*] Quotient Rings
[LIST]
[*] Construction of Quotient Rings
[*] Examples
[*] Fundamental Homomorphism Theorem and Some Consequences
[*] Properties of Prime and Maximal Ideals
[/LIST]
[*] Integral Domains
[LIST]
[*] Characteristic of an Integral Domain
[*] Properties of the Characteristic
[*] Finite Fields
[*] Construction of the Field of Quotients
[/LIST]
[*] The Integers
[LIST]
[*] Ordered Integral Domains
[*] Well-ordering
[*] Characterization of [itex]\mathbb{Z}[/itex] Up to Isomorphism
[*] Mathematical Induction
[*] Division Algorithm
[/LIST]
[*] Factoring Into Primes
[LIST]
[*] Ideals of [itex]\mathbb{Z}[/itex]
[*] Properties of the GCD
[*] Relatively Prime Integers
[*] Primes
[*] Euclid's Lemma
[*] Unique Factorization
[/LIST]
[*] Elements of Number Theory
[LIST]
[*] Properties of Congruence
[*] Theorems of Fermat and Euler
[*] Solutions of Linear Congruences
[*] Chinese Remainder Theorem
[*] Wilson's Theorem and Consequences
[*] Quadratic Residues
[*] The Legendre Symbol
[*] Primitive Roots
[/LIST]
[*] Rings of Polynomials
[LIST]
[*] Motivation and Definitions
[*] Domains of Polynomials over a Field
[*] Division Algorithm
[*] Polynomials in Several Variables
[*] Fields of Polynomial Quotients
[/LIST]
[*] Factoring Polynomials
[LIST]
[*] Ideals of [itex]F[x][/itex]
[*] Properties of the GCD
[*] Irreducible Polynomials
[*] Unique Factorization
[*] Euclidean Algorithm
[/LIST]
[*] Substitution in Polynomials
[LIST]
[*] Roots and Factors
[*] Polynomial Functions
[*] Polynomials over [itex]\mathbb{Q}[/itex]
[*] Eisenstein's Irreducibility Criterion
[*] Polynomials over the Reals
[*] Polynomial Interpolation
[/LIST]
[*] Extensions of Fields
[LIST]
[*] Algebraic and Transcendental Elements
[*] The Minimum Polynomial
[*] Basic Theorem on Field Extensions
[/LIST]
[*] Vector Spaces
[LIST]
[*] Elementary Properties of Vectors Spaces
[*] Linear Independence
[*] Basis
[*] Dimension
[*] Linear Transformations
[/LIST]
[*] Degrees of Field Extensions
[LIST]
[*] Simple and Iterated Extensions
[*] Degree of an Iterated Extension
[*] Field of Algebraic Elements
[*] Algebraic Numbers
[*] Algebraic Closure
[/LIST]
[*] Ruler and Compass
[LIST]
[*] Constructible Points and Numbers
[*] Impossible Constructions
[*] Constructible Angles and Polygons
[/LIST]
[*] Galois Theory: Preamble
[LIST]
[*] Multiple Roots
[*] Root Field
[*] Extension of a Field Isomorphism
[*] Roots of Unity
[*] Separable Polynomials
[*] Normal Extensions
[/LIST]
[*] Galois Theory: The Heart of The Matter
[LIST]
[*] Field Automorphisms
[*] The Galois Group
[*] The Galois Correspondence
[*] Fundamental Theorem of Galois Theory
[*] Computing Galois Groups
[/LIST]
[*] Solving Equations by Radicals
[LIST]
[*] Radical Extensions
[*] Abelian Extensions
[*] Solvable Groups
[*] Insolvability of the Quintic
[/LIST]
[*] Index
[/LIST]
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