From Polynomials to Sums of Squares

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 -- 1 Polynomials in one variable -- 1.1 Polynomials with rational coefficients -- 1.2 Polynomials with coefficients in Zp -- 1.3 Polynomial division -- 1.4 Common divisors of polynomials -- 1.5 Units, irreducibles and the factor theorem -- 1.6 Factorization into irreducible polynomials -- 1.7 Polynomials with integer coefficients -- 1.8 Factorization in Zp [x] and applications to Z[x] -- 1.9 Factorization in Q[x] -- 1.10 Factorizing with the aid of the computer -- Summary of chapter 1 -- Exercises for chapter 1 -- 2 Using polynomials to make new number fields -- 2.1 Roots of irreducible polynomials -- 2.2 The splitting field of xP" - x in Zp [x] -- Summary of chapter 2 -- Exercises for chapter 2 -- 3 Quadratic integers in general and Gaussian integers in particular -- 3.1 Algebraic numbers -- 3.2 Algebraic integers -- 3.3 Quadratic numbers and quadratic integers -- 3.4 The integers of Q(-J=T) -- 3.5 Division with remainder in Z[i] -- 3.6 Prime and composite integers in Z[i] -- Summary of chapter 3 -- Exercises for chapter 3 -- 4 Arithmetic in quadratic domains -- 4.1 Multiplicative norms -- 4.2 Application of norms to units in quadratic domains -- 4.3 Irreducible and prime quadratic integers -- 4.4 Euclidean domains of quadratic integers -- 4.5 Factorization into irreducible integers in quadratic -- domains -- Summary of chapter 4 -- Exercises for chapter 4 -- 5 Composite rational integers and sums of squares -- 5.1 Rational primes -- 5.2 Quadratic residues and the Legendre symbol -- 5.3 Identifying the rational primes that become composite in a quadratic domain -- 5.4 Sums of squares -- Summary of chapter 5 -- Exercises for chapter 5 -- Appendices -- 1 Abstract perspectives -- 1.1 Groups -- 1.2 Rings and integral domains -- 1.3 Divisibility in integral domains -- 1.4 Euclidean domains and factorization into irreducibles -- 1.5 Unique factorization in Euclidean domains -- 1.6 Integral domains and fields -- 1.7 Finite fields -- 2 The product of primitive polynomials -- 3 The Mobius function and cyclotomic polynomials -- 4 Rouches theorem -- 5 Dirichlet's theorem and Pell's equation -- 6 Quadratic reciprocity -- References – Index.