Algebra A book of Abstract Algebra by Pinter

[micromass]

• 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:

[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
[/LIST]
[*] The Definition of Groups
[LIST]
[*] Groups
[*] Examples of Infinite and Finite Groups
[*] Examples of Abelian and Nonabelian Groups
[*] Group Tables
[/LIST]
[*] Elementary Properties of Groups
[LIST]
[*] Uniqueness of Identity and Inverses
[*] Properties of Inverses
[*] Direct Product of Groups
[/LIST]
[*] Subgroups
[LIST]
[*] Definition of Subgroups
[*] Generators and Defining Relations
[*] Cayley Diagrams
[*] Center of a Group
[/LIST]
[*] Functions
[LIST]
[*] Injective, Surjective, Bijective Function
[*] Composite and Inverse of Functions
[/LIST]
[*] Groups of Permutations
[LIST]
[*] Symmetric Groups
[*] Dihedral Groups
[/LIST]
[*] Permutations of a Finite Set
[LIST]
[*] Decomposition of Permutations into Cycles
[*] Transpositions
[*] Even and Odd Permutations
[*] Alternating Groups
[/LIST]
[*] Isomorphism
[LIST]
[*] The Concept of Isomorphism in Mathematics
[*] Isomorphic and Nonisomorphic Groups
[*] Cayley's Theorem
[*] Group Automorphisms
[/LIST]
[*] Order of Groups Elements
[LIST]
[*] Powers/Multiples of Group Elements
[*] Laws of Exponents
[*] Properties of the Order of Group Elements
[/LIST]
[*] Cyclic Groups
[LIST]
[*] Finite and Infinite Cyclic Groups
[*] Isomorphism of Cyclic Groups
[*] Subgroups of Cyclic Groups
[/LIST]
[*] Partitions and Equivalence Relations
[*] Counting Cosets
[LIST]
[*] 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
[/LIST]
[*] Quotient Groups
[LIST]
[*] Quotient Group Construction
[*] Examples and Applications
[*] The Class Equation
[*] Induction on the Order of a Group
[/LIST]
[*] The Fundamental Homomorphism Theorem
[LIST]
[*] 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]

 

[Sankaku]

Pinter's book does what very few mathematics books do. It perfectly balances rigour and intuition. His style is so compelling that parts of the book pull you in like reading a novel. I mainly used it as a supplemental text in my first few abstract algebra courses. However, I credit a large part of my love of the subject to finding Prof. Pinter's book at the right time.

Is it perfect? Not quite. He puts some material in the exercises that probably deserved to be treated in the main part of the text. However, it is unlikely that you will be using this book as your only reference, so I don't feel this is much of a flaw.

Bottom line. For not much more than $10, this belongs in the library of everyone who has any interest in mathematics.

 

[mathwonk]

that certainly does seem well written, at least from what little i can see on amazon. i wish i had known about this book when teaching undergrad courses on proof and intro to algebra.

 

[BloodyFrozen]

Great book. I think a (HS) student with a deep interest in mathematics would enjoy his informal, yet rigorous exposition to the material. If one already knows proofs, a more advanced book would probably be more appropriate.

 

[jbunniii]

I agree with the other reviews. This is a lovely little book, rigorous but extremely well motivated. The main text carries you efficiently from first principles through Galois theory without too many detours. The exercises develop many interesting side topics and examples, with the more difficult material carefully broken down into step by step exercises. I wish I had known about this book when I was first learning algebra. Considering its low price, pretty much everyone should own this book.

 

[jmjlt88]

Pinter's text served as my gateway back into rigorous mathematics! It is well-written and organized. There are a plethora of perfecty chosen problems at the end of each chapter. I highly recommend this text to anyone. In fact, I will be suggesting this text to one of my brighter students for summer reading. (The student will be a senior in high school next year.)