From Polynomials to Sums of Squares

T.H. Jackson

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.