Synopsis
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
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