Algebra Is Algebra by Michael Artin a Good Introduction to Undergraduate-Level Algebra?

[Greg Bernhardt]

• Author: Michael Artin
• Title: Algebra
• Amazon Link: https://www.amazon.com/dp/0132413779/?tag=pfamazon01-20
• Prerequisities: High-school mathematics, proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] A Note for the Teacher
[*] Matrix Operations
[LIST]
[*] The Basic Operations
[*] Row Reduction
[*] Determinants
[*] Permutation Matrices
[*] Cramer's Rule
[*] Exercises
[/LIST]
[*] Groups
[LIST]
[*] The Definition of a Group
[*] Subgroups
[*] Isomorphisms
[*] Homomorphisms
[*] Equivalence Relations and Partitions
[*] Cosets
[*] Restriction of a Homomorphism to a Subgroup
[*] Products of Groups
[*] Modular Arithmetic
[*] Quotient Groups
[*] Exercises
[/LIST]
[*] Vector Spaces
[LIST]
[*] Real Vector Spaces
[*] Abstract Fields
[*] Bases and Dimension
[*] Computation with Bases
[*] Infinite-Dimensional Spaces
[*] Direct Sums
[*] Exercises
[/LIST]
[*] Linear Transformations
[LIST]
[*] The Dimension Formula
[*] The Matrix of a Linear Transformation
[*] Linear Operators and Eigenvectors
[*] The Characteristic Polynomial
[*] Orthogonal Matrices and Rotations
[*] Diagonalization
[*] Systems of Differential Equations
[*] The Matrix Exponential
[*] Exercises
[/LIST]
[*] Symmetry
[LIST]
[*] Symmetry of Plane Figures
[*] The Group of Motions of the Plane
[*] Finite Groups of Motions
[*] Discrete Groups of Motions
[*] Abstract Symmetry: Group Operations
[*] The Operation on Cosets
[*] The Counting Formula
[*] Permutation Representations
[*] Finite Subgroups of the Rotation Group
[*] Exercises
[/LIST]
[*] More Group Theory
[LIST]
[*] The Operations of a Group on Itself
[*] The Class Equation of the Icosahedral Group
[*] Operations on Subsets
[*] The Sylow Theorems
[*] The Groups of Order 12
[*] Computation in the Symmetric Group
[*] The Free Group
[*] Generators and Relations
[*] The Todd-Coxeter Algorithm
[*] Exercises
[/LIST]
[*] Bilinear Forms
[LIST]
[*] Definition of Bilinear Form
[*] Symmetric Forms: Orthogonality
[*] The Geometry Associated to a Positive Form
[*] Hermitian Forms
[*] The Spectral Theorem
[*] Conics and Quadrics
[*] The Spectral Theorem for Normal Operators
[*] Skew-Symmetric Forms
[*] Summary of Results, in Matrix Notation
[*] Exercises
[/LIST]
[*] Linear Groups
[LIST]
[*] The Classical Linear Groups
[*] The Special Unitary Group [itex]SU_2[/itex]
[*] The Orthogonal Representation of [itex]SU_2[/itex]
[*] The Special Linear Group [itex]SL_2(\mathbb{R})[/itex]
[*] One-Parameter Subgroups
[*] The Lie Algebra
[*] Translation in a Group
[*] Simple Groups
[*] Exercises
[/LIST]
[*] Group Representations
[LIST]
[*] Definition of a Group Representation
[*] G-Invariant Forms and Unitary Representations
[*] Compact Groups
[*] G-Invariant Subspaces and Irreducible Representations
[*] Characters
[*] Permutation Representations and the Regular Representation
[*] The Representations of the Icosahedral Group
[*] One-Dimensional Representations
[*] Schur's Lemma, and Proof of the Orthogonality Relations
[*] Representations of the Group [itex]SU_2[/itex]
[*] Exercises
[/LIST]
[*] Rings
[LIST]
[*] Definition of a Ring
[*] Formal Construction of Integers and Polynomials
[*] Homomorphisms and Ideals
[*] Quotient Rings and Relations in a Ring
[*] Adjunction of Elements
[*] Integral Domains and Fraction Fields
[*] Maximal Ideals
[*] Algebraic Geometry
[*] Exercises
[/LIST]
[*] Factorization
[LIST]
[*] Factorization of Integers and Polynomials
[*] Unique Factorization Domains, Principal Ideal Domains, and Euclidean Domains
[*] Gauss's Lemma
[*] Explicit Factorization of Polynomials
[*] Primes in the Ring of Gauss Integers
[*] Algebraic Integers
[*] Factorization in Imaginary Quadratic Fields
[*] Ideal Factorization
[*] The Relation Between Prime Ideals of R and Prime Integers
[*] Ideal Classes in Imaginary Quadratic Fields
[*] Real Quadratic Fields
[*] Some Diophantine Equations
[*] Exercises
[/LIST]
[*] Modules
[LIST]
[*] The Definition of a Module
[*] Matrices, Free Modules, and Bases
[*] The Principle of Permanence of Identities
[*] Diagonalization of Integer Matrices
[*] Generators and Relations for Modules
[*] The Structure Theorem for Abelian Groups
[*] Application to Linear Operators
[*] Free Modules over Polynomial Rings
[*] Exercises
[/LIST]
[*] Fields
[LIST]
[*] Examples of Fields
[*] Algebraic and Transcendental Elements
[*] The Degree of a Field Extension
[*] Constructions with Ruler and Compass
[*] Symbolic Adjunction of Roots
[*] Finite Fields
[*] Function Fields
[*] Transcendental Extensions
[*] Algebraically Closed Fields
[*] Exercises
[/LIST]
[*] Galois Theory
[LIST]
[*] The Main Theorem of Galois Theory
[*] Cubic Equations
[*] Symmetric Functions
[*] Primitive Elements
[*] Proof of the Main Theorem
[*] Quartic Equations
[*] Kummer Extensions
[*] Cyclotomic Extensions
[*] Quintic Equations
[*] Exercises
[/LIST]
[*] Appendix: Background Material 
[LIST]
[*] Set Theory
[*] Techniques of Proof
[*] Topology
[*] The Implicit Function Theorem
[*] Exercises
[/LIST]
[*] Notation 
[*] Suggestions for Further Reading 
[*] Index 
[/LIST]

 

[micromass]

Artin is a top notch mathematician and this is very apparent from this book. The book treats the basics of abstract algebra in a really nice way. Furthermore, there are some nice additions such as symmetry of plane figures. If you want to start studying abstract algebra and you're looking for a nice first book, then this is the ideal book for you. Don't expect the book to be easy though. A course on proofs and logic seems necessary before doing this book.

 

[mathwonk]

to set the expectations and level, this was the sophomore level book at MIT. Those of us who would not even get into MIT thus may expect it to be hard. But it will repay our efforts. I have also taught from it at UGA. Mike's proofs are really clear, no hand waving and no cribbing from other books. he explains everything as he sees it himself, trying to make it come alive for the reader. he always sticks to the same level of exposition too, for (smart hard working) beginners.

 

[A. Bahat]

This is where I first learned algebra. It is an excellent book, written in an "organic" style reminiscent of Arnold, Atiyah, Poincare, Riemann, etc.--the theory is always well-motivated, and abstraction for abstraction's sake is kept at bay. There are important topics not covered in Artin (dual spaces and multilinear algebra, for example), so you will need to go beyond Artin at some point. But for its intended purpose, an introduction to algebra, I can think of no better choice.