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Algebra (2nd Edition) by Michael Artin



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Algebra (2nd Edition) written by Michael Artin , Massachusetts Institute of Technology.
According to Michael Artin:
This book began many years ago in the form of supplementary notes for my algebra classes. I wanted to discuss someconcretetopics such as symmetry, linear groups, and quadratic number fieldsin more detail than the text provided, and to shift the emphasis in group theory from permutation groupsto matrix groups. J...Jattices, another recurring theme, appeared spontaneously.
My hope was that the concrete material would interest the students and that it would make the abstractions more understandable- in short, that they could get farther bylearning both at the same time. This worked pretty welLIt tookmequite a while to decide what to include, but I gradually handed out more notesand eventually began teaching from them without another text. Though this produced a bookthat is different from most others, the problemsI encountered while fitting the parts together caused me many headaches. I can't recommend the method.

Algebra (2nd Edition) written by Michael Artin cover the following topics.


  • 1. Matrices
    1.1 The Basic Operations
    1.2 Row Reduction
    1.3 The Matrix Transpose
    1.4 Determinants
    1.5 Permutations
    1.6 Other Formulas for the Determinant
    Exercises

  • 2. Groups
    2.1 Lawsof Composition
    2.2 Groups and Subgroups
    2.3 Subgroups of the Additive Group of Integers
    2.4 Cyclic Groups
    2.5 Homomorphisms
    2.6 Isomorphisms
    2.7 Equivalence Relations and Partitions
    2.8 Cosets
    2.9 Modular Arithmetic
    2.10 The Correspondence Theorem
    2.11 Product Groups
    2.12 Quotient Groups
    Exercises

  • 3. Vector Spaces
    3.1 Subspacesof }Rn
    3.2 Fields
    3.3 Vector Spaces
    3.4 Bases and Dimension
    3.5 Computing with Bases
    3.6 Direct Sums
    3.7 Infinite-Dimensional Spaces
    Exercises

  • 4. Linear Operators
    4.1 The Dimension Formula
    4.2 The Matrix of a Linear Transformation)
    4.3 Linear Operators
    4.4 Eigenvectors
    4.5 The Characteristic Polynomial
    4.6 Triangular and Diagonal Forms
    4.7 JordanForm
    Exercises

  • 5. Applications of LinearOperators
    5.1 Orthogonal Matrices and Rotations
    5.2 Using Continuity
    5.3 Systems of Differential Equations
    5.4 The Matrix Exponential
    Exercises

  • 6. Symmetry
    6.1 Symmetry of Plane Figures
    6.2 Isometries
    6.3 Isometries of the Plane
    6.4 Finite Groups of Orthogonal Operators on the Plane
    6.5 DiscreteGroups of Isometries
    6.6 Plane Crystallographic Groups
    6.7 Abstract Symmetry: Group Operations
    6.8 TheOperation on Cosets . . . .
    6.9 The Counting Formula ....
    6.10 Operations on Subsets . . . .. ....
    6.11 Permutation Representations . . . . . . . . . .
    6.12 Finite Subgroups of the Rotation Group Exercises . . . . . . . . . . . . . . . . . .)

  • 7. More Group Theory
    7.1 Cayley's Theorem
    7.2 The Class Equation
    7.3 p-Groups
    7.4 The Class Equation of the Icosahedral Group
    7.5 Conjugation in the Symmetric Group
    7.6 Normalizers
    7.7 The Sylow Theorems
    7.8 Groups of Order 12
    7.9 The Free Group
    7.10 Generators and Relations
    7.11 The Todd-Coxeter Algorithm
    Exercises

  • 8. Bilinear Forms
    8.1 Bilinear Forms
    8.2 Symmetric Forms
    8.3 Hermitian Forms
    8.4 Orthogonality
    8.5 Euclidean Spacesand Hermitian Spaces
    8.6 The Spectral Theorem
    8.7 Conics and Quadrics
    8.8 Skew-Symmetric Forms
    8.9 Summary
    Exercises

  • 9. Linear Groups
    9.1 The ClassicalGroups
    9.2 Interlude: Spheres
    9.3 The Special Unitary Group SU2
    9.4 The Rotation Group S03
    9.5 One-Parameter Groups
    9.6 The Lie Algebra
    9.7 Translation in a Group
    9.8 Normal Subgroups of SL2
    Exercises

  • 10. Group Representations
    10.1 Definitions
    10.2 Irreducible Representations
    10.3 U ni tary Represen tations
    10.4 Characters
    10.5 One-DimensionalCharacters
    10.6 TheRegularRepresentation
    10.7 Schur
    s Lemma
    10.8 Proof of the Orthogonality Relations
    10.9 Representations of SU2
    Exercises

  • 11. Rings
    11.1 Definition of a Ring
    11.2 Polynomial Rings
    11.3 Homomorphisms and Ideals
    11.4Quotient Rings
    11.5 Adjoining Elements
    11.6 Product Rings
    11.7 Fractions
    11.8Maximal Ideals
    11.9 Algebraic Geometry
    Exercises

  • 12. Factoring
    12.1 Factoring Integers
    12.2 Unique Factorization Domains
    12.3 Gauss
    8 Lemma
    12.4 Factoring Integer Polynomials
    12.5 Gauss Primes
    Exercises

  • 13. Quadratic Number Fields
    13.1 Algebraic Integers
    13.2 Factoring Algebraic Integers
    13.3 Ideals in Z[H ]
    13.4 Ideal Multiplication
    13.5 Factoring Ideals
    13.6 Prime Ideals and Prime Integers
    13.7 Ideal Classes
    13.8 Computing the Class Group
    13.9 Real Quadratic Fields
    13.1 0 About Lattices
    Exercises

  • 14. Linear Algebra in a Ring
    14.1 Modules
    14.2Free Modules
    14.3 Identi ti es
    14.4 Diagonalizing Integer Matrices
    14.5 Generators and Relations
    14.6 Noetherian Rings
    14.7 Structure of Abelian Groups
    14.8 Application to Linear Operators
    14.9 Polynomial Rings in Several Variables
    Exercises

  • 15. Fields
    15.1 Examples of Fields
    15.2 Algebraic and Transcendental Elements
    15.3The Degree of a Field Extension
    15.4 Finding the Irreducible Polynomial
    15.5Ruler and Compass Constructions
    15..6 Adjoining Roots
    15.7 Finite Fields
    15.8 Primitive Elements
    15.9 Function Fields
    15.10 The Fundamental Theorem of Algebra
    Exercises

  • 16. Galois Theory
    16.1 SymmetricFunctions
    16.2 The Discriminant
    16.3 Splitting Fields
    16.4 Isomorphisms of Field Extensions
    16.5 Fixed Fields
    16.6 Galois Extensions
    16.7 The Main Theorem
    16.8 Cubic Equations
    16.9 Quartic Equations
    16.10 Roots of Unity
    16.11 Kummer Extensions
    16.12 Quintic Equations
    Exercises

  • APPENDIX (Background Material)
    A.l About Proofs
    A.2 The Integers
    A.3 Zorn's Lemma
    A.4 TheImplicit Function Theorem
    Exercises

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