The theory of polynomials constitutes an essential part of university of algebra and calculus. Nevertheless, there are very few books entirely devoted to this theory. This book provides an exposition of the main results in the theory of polynomials, both classical and modern. Considerable attention is given to Hilbert's 17th problem on the representation of non-negative polynomials by the sums of squares of rational functions and its generalizations. Many of the modern results have only been published in journals so far.

Covers its topic in greater depth than the typical standard books on polynomial algebra

Foreword Notational conventions Chapter 1. Roots of polynomials 1. Inequalities for roots 2. The roots of a polynomial and of its derivative 3. The resultant and the discriminant 4. Separation of roots 5. Lagrange's series and estimates of the roots of a polynomial 6. Problems to Chapter 1 7. Solutions of selected problems Chapter 2. Irreducible polynomials 1. Main properties of irreducible polynomials 2. Irreducibility criteria 3. Irreducibility of trinomials and fournomials 4. Hilbert's irreducibility theorem 5. Algorithms for factorization into irreducible factors 6. Problems to Chapter 2 7. Solutions of selected problems Chapter 3. Polynomials of a particular form 1. Symmetric polynomials 2. Integer-valued polynomials 3. Cyclotomic polynomials 4. Chebyshev polynomials 5. Bernoulli's polynomials 6. Problems to Chapter 3 7. Solutions of selected problems Chapter 4. Certain properties of polynomials 1. Polynomials with prescribed values 2. The height of a polynomial and other norms 3. Equations for polynomials 4. Transformations of polynomials 5. Algebraic numbers 6. Problems to Chapter 4 Chapter 5. Galois theory 1. Lagrange's theorem and the Galois resolvent 2. Basic Galois theory 3. How to solve equations by radicals 4. Calculations of the Galois groups Chapter 6. Ideals in polynomial rings 1. Hilbert's basis theorem and Hilbert's theorem on zeros 2. Gröbner bases Chapter 7. Hilbert's seventeenth problem 1. The sums of squares: introduction 2. Artin's theory 3. Pfister's theory Chapter 8. Appendix 1. The Lenstra-Lenstra-Lovasz algorithm Bibliography