Description
Springer Polynomials by Victor V. Prasolov, Dimitry Leites
Covers its topic in greater depth than the typical standard books on polynomial algebra_x000D_ Table of contents :- _x000D_
Foreword_x000D_
Notational conventions_x000D_
Chapter 1. Roots of polynomials_x000D_
1. Inequalities for roots_x000D_
2. The roots of a polynomial and of its derivative_x000D_
3. The resultant and the discriminant_x000D_
4. Separation of roots_x000D_
5. Lagrange's series and estimates of the roots of a polynomial_x000D_
6. Problems to Chapter 1_x000D_
7. Solutions of selected problems_x000D_
Chapter 2. Irreducible polynomials_x000D_
1. Main properties of irreducible polynomials_x000D_
2. Irreducibility criteria_x000D_
3. Irreducibility of trinomials and fournomials_x000D_
4. Hilbert's irreducibility theorem_x000D_
5. Algorithms for factorization into irreducible factors_x000D_
6. Problems to Chapter 2_x000D_
7. Solutions of selected problems_x000D_
Chapter 3. Polynomials of a particular form_x000D_
1. Symmetric polynomials_x000D_
2. Integer-valued polynomials_x000D_
3. Cyclotomic polynomials_x000D_
4. Chebyshev polynomials_x000D_
5. Bernoulli's polynomials_x000D_
6. Problems to Chapter 3_x000D_
7. Solutions of selected problems_x000D_
Chapter 4. Certain properties of polynomials_x000D_
1. Polynomials with prescribed values_x000D_
2. The height of a polynomial and other norms_x000D_
3. Equations for polynomials_x000D_
4. Transformations of polynomials_x000D_
5. Algebraic numbers_x000D_
6. Problems to Chapter 4_x000D_
Chapter 5. Galois theory_x000D_
1. Lagrange's theorem and the Galois resolvent_x000D_
2. Basic Galois theory_x000D_
3. How to solve equations by radicals_x000D_
4. Calculations of the Galois groups_x000D_
Chapter 6. Ideals in polynomial rings_x000D_
1. Hilbert's basis theorem and Hilbert's theorem on zeros_x000D_
2. Groebner bases_x000D_
Chapter 7. Hilbert's seventeenth problem_x000D_
1. The sums of squares: introduction_x000D_
2. Artin's theory_x000D_
3. Pfister's theory_x000D_
Chapter 8. Appendix_x000D_
1. The Lenstra-Lenstra-Lovasz algorithm_x000D_
Bibliography_x000D_