Polynomials

ForewordNotational conventionsChapter 1. Roots of polynomials1. Inequalities for roots2. The roots of a polynomial and of its derivative3. The resultant and the discriminant4. Separation of roots5. Lagrange's series and estimates of the roots of a polynomial6. Problems to Chapter 17. Solutions of selected problemsChapter 2. Irreducible polynomials1. Main properties of irreducible polynomials2. Irreducibility criteria3. Irreducibility of trinomials and fournomials4. Hilbert's irreducibility theorem5. Algorithms for factorization into irreducible factors6. Problems to Chapter 27. Solutions of selected problemsChapter 3. Polynomials of a particular form1. Symmetric polynomials2. Integer-valued polynomials3. Cyclotomic polynomials4. Chebyshev polynomials5. Bernoulli's polynomials6. Problems to Chapter 37. Solutions of selected problemsChapter 4. Certain properties of polynomials1. Polynomials with prescribed values2. The height of a polynomial and other norms3. Equations for polynomials4. Transformations of polynomials5. Algebraic numbers6. Problems to Chapter 4Chapter 5. Galois theory1. Lagrange's theorem and the Galois resolvent2. Basic Galois theory3. How to solve equations by radicals4. Calculations of the Galois groupsChapter 6. Ideals in polynomial rings1. Hilbert's basis theorem and Hilbert's theorem on zeros2. Gröbner basesChapter 7. Hilbert's seventeenth problem1. The sums of squares: introduction2. Artin's theory3. Pfister's theoryChapter 8. Appendix1. The Lenstra-Lenstra-Lovasz algorithmBibliography