1st Edition

# From Polynomials to Sums of Squares

194 Pages
by CRC Press

194 Pages
by CRC Press

194 Pages
by CRC Press

Also available as eBook on:

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

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
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

Multiplicative norms
Application of norms to units in quadratic domains
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