1st Edition

From Polynomials to Sums of Squares

By T.H Jackson Copyright 1995
    194 Pages
    by CRC Press

    194 Pages
    by CRC Press

    From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.

    Preface
    Software Copyright and Site License

    POLYNOMIALS IN ONE VARIABLE
    Polynomials with rational coefficients
    Polynomials with coefficients in Z^O^Ip
    Polynomial division
    Common divisors of polynomials
    Units, irreducibles and the factor theorem
    Factorization into irreducible polynomials
    Polynomials with integer coefficients
    Factorization in Z^O^Ip[^Ix] and applications to Z[^Ix]
    Factorization in Q[^Ix]
    Factorizing with the aid of the computer
    Summary of chapter 1
    Exercises for chapter 1

    USING POLYNOMIALS TO MAKE NEW NUMBER FIELDS
    Roots of irreducible polynomials
    The splitting field of ^Ix^Tpn - ^Ix in Z^O^Ip[^Ix]
    Summary of chapter 2
    Exercises for chapter 2

    QUADRATIC INTEGERS IN GENERAL AND GAUSSIAN INTEGERS IN PARTICULAR
    Algebraic numbers
    Algebraic integers
    Quadratic numbers and quadratic integers
    The integers of Q((square root) -1)
    Division with remainder in Z[^Ii]
    Prime and composite integers in Z[^Ii]
    Summary of chapter 3
    Exercises for chapter 3

    ARITHMETIC IN QUADRATIC DOMAINS
    Multiplicative norms
    Application of norms to units in quadratic domains
    Irreducible and prime quadratic integers
    Euclidean domains of quadratic integers
    Factorization into irreducible integers in quadratic domains
    Summary of chapter 4
    Exercises for chapter 4

    COMPOSITE RATIONAL INTEGERS AND SUMS OF SQUARES
    Rational primes
    Quadratic residues and the Legendre symbol
    Identifying the rational primes that become composite in a quadratic domain
    Sums of squares
    Summary of chapter 5
    Exercises for chapter 5

    APPENDICES
    Abstract perspectives
    Groups
    Rings and integral domains
    Divisibility in integral domains
    Euclidean domains and factorization into irreducibles
    Unique factorization in Euclidean domains
    Integral domains and fields
    Finite fields
    The product of primitive polynomials
    The M^D"obius function and cyclotomic polynomials
    Rouch^D'es theorem
    Dirichlet's theorem and Pell's equation
    Quadratic reciprocity

    REFERENCES
    INDEX

    Biography

    T.H. Jackson