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Algebra by Michael Artin



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Algebra written by Michael Artin , Massachusetts Institute of Technology.
According to Michael Artin:
This book began about 20 years ago in the form of supplementary notes for my algebra classes. I wanted to discuss some concrete topics such as symmetry, linear groups, and quadratic number fields in more detail than the text provided, and to shift the emphasis in group theory from permutation groups to matrix groups. Lattices, 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 by learning both at the same time. This worked pretty well. It took me quite a while to decide what I wanted to put in, but I gradually handed out more notes and eventually began teaching from them without another text. This method produced a book which is, I think, somewhat different from existing ones. However, the problems I encountered while fitting the parts together caused me many headaches, so I can't recommend starting this way.
The main novel feature of the book is its increased emphasis on special topics. They tended to expand each time the sections were rewritten, because I noticed over the years that, with concrete mathematics in contrast to abstract concepts, students often prefer more to less. As a result, the ones mentioned above have become major parts of the book. There are also several unusual short subjects, such as the ToddCoxeter algorithm and the simplicity of PSL2•

Algebra written by Michael Artin cover the following topics.

  • 1. Matrix Operations
    1. The Basic Operations
    2. Row Reduction
    3. Determinants
    4. Permutation Matrices
    5. Cramer
    s Rule
    EXERCISES

  • 2. Groups
    1. The Definition of a Group
    2. Subgroups
    3. Isomorphisms
    4. Homomorphisms
    s.Equivalence Relations And Partitions
    6. Cosets
    7. Restriction of a Homomorphism to a Subgroup
    8. Products of Groups
    9. Modular Arithmetic
    10. Quotient Groups
    EXERCISES

  • 3. Vector Spaces
    1. Real Vector Spaces
    2. Abstract Fields
    3. Bases and Dimension
    4. Computation with Bases
    5. Infinite-Dimensional Spaces
    6. Direct Sums
    EXERCISES

  • 4. Linear Transformations
    1. The Dimension Formula
    2. The Matrix of a Linear Transformation III
    3. Linear Operators and Eigenvectors
    4. The Characteristic Polynomial
    5. Orthogonal Matrices and Rotations
    6. Diagonalization
    7. Systems of Differential Equations
    8. The Matrix Exponential
    EXERCISES

  • 5. Symmetry
    1. Symmetry of Plane Figures
    2. The Group of Motions of the Plane
    3. Finite Groups of Motions
    4. Discrete Groups of Motions
    5. Abstract Symmetry: Group Operations
    6. The Operation on Cosets
    7. The Counting Formula
    8. Permutation Representations
    9. Finite Subgroups of the Rotation Group
    EXERCISES

  • 6. More Group Theory
    1. The Operations of a Group on Itself
    2. The Class Equation of the Icosahedral Group
    3. Operations on Subsets
    4. The Sylow Theorems
    5. The Groups of Order 12
    6. Computation in the Symmetric Group
    7. The Free Group
    8. Generators and Relations
    9. The Todd-Coxeter Algorithm
    EXERCISES

  • 7. Bilinear Forms
    1. Definition of Bilinear Form
    2. Symmetric Forms: Orthogonality
    3. The Geometry Associated to a Positive Form
    4. Hermitian Forms
    5. The Spectral Theorem
    6. Conics and Quadrics
    7. The Spectral Theorem for Normal Operators
    8. Skew-Symmetric Forms
    9. Summary of Results, in Matrix Notation
    EXERCISES

  • 8. Linear Groups
    1. The Classical Linear Groups
    2. The Special Unitary Group SU2
    3. The Orthogonal Representation of SU2
    4. The Special Linear Group SL2(lR)
    5. One-Parameter Subgroups
    6. The Lie Algebra
    7. Translation in a Group
    8. Simple Groups
    EXERCISES

  • 9. Group Representations
    1. Definition of a Group Representation
    2. G-Invariant Forms and Unitary Representations
    3. Compact Groups
    4. G-Invariant Subspaces and Irreducible Representations
    5. Characters
    6. Permutation Representations and the Regular Representation
    7. The Representations of the Icosahedral Group
    8. One-Dimensional Representations
    9. Schur
    s Lemma, and Proof of the Orthogonality Relations
    10. Representations of the Group SU2
    EXERCISES

  • 10. Rings
    1. Definition of a Ring
    2. Formal Construction of Integers and Polynomials
    3. Homomorphisms and Ideals
    4. Quotient Rings and Relations in a Ring
    5. Adjunction of Elements
    6. Integral Domains and Fraction Fields
    7. Maximal Ideals
    8. Algebraic Geometry
    EXERCISES

  • 11. Factorization
    1. Factorization of Integers and Polynomials
    2. Unique Factorization Domains, Principal Ideal Domains, and Euclidean Domains
    3. Gauss
    s Lemma
    4. Explicit Factorization of Polynomials
    5. Primes in the Ring of Gauss Integers
    6. Algebraic Integers
    7. Factorization in Imaginary Quadratic Fields
    8. Ideal Factorization
    9. The Relation Between Prime Ideals of R and Prime Integers
    10. Ideal Classes in Imaginary Quadratic Fields
    11. Real Quadratic Fields
    12. Some Diophantine Equations
    EXERCISES

  • 12. Modules
    1. The Definition of a Module
    2. Matrices, Free Modules, and Bases
    3. The Principle of Permanence of Identities
    4. Diagonalization of Integer Matrices
    5. Generators and Relations for Modules
    6. The Structure Theorem for Abelian Groups
    7. Application to Linear Operators
    8. Free Modules over Polynomial Rings
    EXERCISES

  • 13. Fields
    1. Examples of Fields
    2. Algebraic and Transcendental Elements
    3. The Degree of a Field Extension
    4. Constructions with Ruler and Compass
    5. Symbolic Adjunction of Roots
    6. Finite Fields
    7. Function Fields
    8. Transcendental Extensions
    9. Algebraically Closed Fields
    EXERCISES

  • 14. Galois Theory
    1. The Main Theorem of Galois Theory
    2. Cubic Equations
    3. Symmetric Functions
    4. Primitive Elements
    5. Proof of the Main Theorem
    6. Quartic Equations
    7. Kummer Extensions
    8. Cyclotomic Extensions
    9. Quintic Equations
    EXERCISES

  • Appendix Background Material
    1. Set Theory
    2. Techniques of Proof
    3. Topology
    4. The Implicit Function Theorem
    EXERCISES

  • Notation
    Suggestions for Further Reading
    Index

  • Start Now
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