1st Edition

Algebra A Computational Introduction

By John Scherk Copyright 2000
    336 Pages
    by Chapman & Hall

    Adequate texts that introduce the concepts of abstract algebra are plentiful. None, however, are more suited to those needing a mathematical background for careers in engineering, computer science, the physical sciences, industry, or finance than Algebra: A Computational Introduction. Along with a unique approach and presentation, the author demonstrates how software can be used as a problem-solving tool for algebra.

    A variety of factors set this text apart. Its clear exposition, with each chapter building upon the previous ones, provides greater clarity for the reader. The author first introduces permutation groups, then linear groups, before finally tackling abstract groups. He carefully motivates Galois theory by introducing Galois groups as symmetry groups. He includes many computations, both as examples and as exercises. All of this works to better prepare readers for understanding the more abstract concepts.

    By carefully integrating the use of Mathematica® throughout the book in examples and exercises, the author helps readers develop a deeper understanding and appreciation of the material. The numerous exercises and examples along with downloads available from the Internet help establish a valuable working knowledge of Mathematica and provide a good reference for complex problems encountered in the field.

    CONGRUENCES
    Basic Properties
    Divisibility Tests
    Common Divisors
    Solving Congruences
    The Integers Modulo n
    Introduction to Software
    PERMUTATIONS
    Permutations as Mappings
    Cycles
    Sign of a Permutation
    PERMUTATION GROUPS
    Definition
    Cyclic Groups
    Generators
    Software and Calculations
    LINEAR GROUPS
    Definitions and Examples
    Generators
    Software and Calculations
    GROUPS
    Basic Properties and More Examples
    Homomorphisms
    SUBGROUPS
    Definition
    Orthogonal Groups
    Cyclic Subgroups and Generators
    Kernel and Image of a Homomorphism
    SYMMETRY GROUPS
    Symmetries of Regular Polygons
    Symmetries of Platonic Solids
    Improper Symmetries
    Symmetries of Equations
    GROUP ACTIONS
    Examples
    Orbits and Stabilizers
    Fractional Linear Transformations
    Cayley's Theorem
    Software and Calculations
    COUNTING FORMULAS
    The Class Equation
    A First Application
    Burnside's Counting Lemma
    Finite Subgroups of SO(3)
    COSETS
    Lagrange's Theorem
    Normal Subgroups
    Quotient Groups
    The Canonical Isomorphism
    Software and Calculations
    SYLOW SUBGROUPS
    The Sylow Theorems
    Groups of Small Order
    A List
    A Calculation
    SIMPLE GROUPS
    Composition Series
    Simplicity of An
    Simplicity of PSL(2,Fp)
    ABELIAN GROUPS
    Free Abelian Groups
    Row and Column Reduction of Integer Matrices
    Classification Theorems
    Invariance of Elementary Divisors
    The Multiplicative Group of the Integers Mod n
    POLYNOMIAL RINGS
    Basic Properties of Polynomials
    Unique Factorization into Irreducibles
    Finding Irreducible Polynomials
    Commutative Rings
    Congruences
    Factoring Polynomials over a Finite Field
    Calculations
    SYMMETRIC POLYNOMIALS
    Polynomials in Several Variables
    Symmetric Polynomials and Functions
    Sums of Powers
    Discriminants
    Software
    ROOTS OF EQUATIONS
    Introduction
    Extension Fields
    Degree of an Extension
    Splitting Fields
    Cubics
    Cyclotomic Polynomials
    Finite Fields
    Plots and Calculations
    GALOIS GROUPS
    Introduction
    Definition
    How Large is the Galois Group?
    The Galois Correspondence
    Discriminants
    QUARTICS
    Galois Groups of Quartics
    The Geometry of the Cubic Resolvent
    Software
    THE GENERAL EQUATION OF THE nth DEGREE
    Examples
    Symmetric Functions
    The Fundamental Theorem of Algebra
    SOLUTION BY RADICALS
    Formulas for a Cubic
    Cyclic Extensions
    Solution by Radicals in Higher Degrees
    Calculations
    RULER-AND-COMPASS CONSTRUCTIONS
    Introduction
    Algebraic Interpretation
    Construction of Regular Polygons
    Periods
    APPENDIX: MATHEMATICA COMMANDS

    Biography

    John Scherk

    "… emphasizes the computational aspects of modern abstract algebra…author has integrated the software Mathematica into the discussions-especially in the group theory sections-but is careful not to make any logical reliance on this software. For one wishing to see the theory unfold through a highly computational approach, this text has much to recommend …writing is logical but not excessively formal…I feel that this text was very courageously written…[the] focus is a bit more narrow that that of the typical first-year undergraduate course in abstract algebra. Yet, if one wishes to develop a deep and intuitive rapport with basic group and Galois theory, then this text has much to offer."
    --David B. Surowski, in Mathematical Reviews, Issue 2001i