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algebra: abstract and concrete: 2.6 goodman frederick [pdf]

Algebra Abstract and Concrete Edition 2.6 by Fredrick M. Goodman



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Algebra Abstract and Concrete Edition 2.6 written by Fredrick M. Goodman. This introduction to modern or abstract algebra addresses the conventional topics of groups, rings, and fields with symmetry as a unifying theme, while it introduces readers to the active practice of mathematics. Its accessible presentation is designed to teach users to think things through for themselves and change their view of mathematics from a system of rules and procedures, to an arena of inquiry. The volume provides plentiful exercises that give users the opportunity to participate and investigate algebraic and geometric ideas which are interesting, important, and worth thinking about. The volume addresses algebraic themes, basic theory of groups and products of groups, symmetries of polyhedra, actions of groups, rings, field extensions, and solvability and isometry groups. For those interested in a concrete presentation of abstract algebra.

Algebra Abstract and Concrete Edition 2.6 written by Fredrick M. Goodman cover the following topics.

  • Chapter 1. Algebraic Themes
    What Is Symmetry?, Symmetries of the Rectangle and the Square, Multiplication Tables, Symmetries and Matrices, Permutations, Divisibility in the Integers, Modular Arithmetic, Polynomials, Counting, Groups, Rings and Fields, An Application to Cryptography

  • Chapter 2. Basic Theory of Groups
    First Results, Subgroups and Cyclic Groups, The Dihedral Groups, Homomorphisms and Isomorphisms, Cosets and Lagrange’s Theorem, Equivalence Relations and Set Partitions, Quotient Groups and Homomorphism Theorems

  • Chapter 3. Products of Groups
    Direct Products, Semidirect Products, Vector Spaces, The dual of a vector space and matrices, Linear algebra over Z, Finitely generated abelian groups

  • Chapter 4. Symmetries of Polyhedra
    Rotations of Regular Polyhedra, Rotations of the Dodecahedron and Icosahedron, What about Reflections?, Linear Isometries, The Full Symmetry Group and Chirality

  • Chapter 5. Actions of Groups
    Group Actions on Sets, Group Actions—Counting Orbits, Symmetries of Groups, Group Actions and Group Structure, Application: Transitive Subgroups of S5, Additional Exercises for Chapter 5

  • Chapter 6. Rings
    A Recollection of Rings, Homomorphisms and Ideals, Quotient Rings, Integral Domains, Euclidean Domains, Principal Ideal, Domains, and Unique Factorization, Unique Factorization Domains, Noetherian Rings, Irreducibility Criteria

  • Chapter 7. Field Extensions – First Look
    A Brief History, Solving the Cubic Equation, Adjoining Algebraic Elements to a Field, Splitting Field of a Cubic Polynomial, Splitting Fields of Polynomials in CŒx

  • Chapter 8. Modules
    The idea of a module, Homomorphisms and quotient modules, Multilinear maps and determinants, Finitely generated Modules over a PID, part I, Finitely generated Modules over a PID, part II., Rational canonical form, Jordan Canonical Form

  • Chapter 9. Field Extensions – Second Look
    Finite and Algebraic Extensions, Splitting Fields, The Derivative and Multiple Roots, Splitting Fields and Automorphisms, The Galois Correspondence, Symmetric Functions, The Derivative and Multiple Roots, Splitting Fields and Automorphisms, The Galois Correspondence, Symmetric Functions, The General Equation of Degree n, Quartic Polynomials, Galois Groups of Higher Degree Polynomials

  • Chapter 10. Solvability
    Composition Series and Solvable Groups, Commutators and Solvability, Simplicity of the Alternating Groups, Cyclotomic Polynomials, The Equation x^n - b, Solvability by Radicals, Radical Extensions

  • Chapter 11. Isometry Groups
    More on Isometries of Euclidean Space, Euler’s Theorem, Finite Rotation Groups, Crystals

  • Appendix A. Almost Enough about Logic
    Statements, Logical Connectives, Quantifiers, Deductions

  • Appendix B. Almost Enough about Sets
    Families of Sets; Unions and Intersections, Finite and Infinite Sets

  • Appendix C. Induction
    Proof by Induction, Definitions by Induction, Multiple Induction

  • Appendix D. Complex Numbers
  • Appendix E. Review of Linear Algebra
    Linear algebra in Kn, Bases and Dimension, Inner Product and Orthonormal Bases

  • Appendix G. Suggestions for Further Study
  • Appendix F. Models of Regular Polyhedra
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