E-raamat: From Polynomials to Sums of Squares

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.

Jackson, T.H