From Polynomials to Sums of Squares: 1st Edition (Paperback) book cover

From Polynomials to Sums of Squares

1st Edition

By T.H Jackson

CRC Press

196 pages

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Paperback: 9780750303293
pub: 1995-01-01
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Description

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.

Table of Contents

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

Subject Categories

BISAC Subject Codes/Headings:
MAT022000
MATHEMATICS / Number Theory