Polynomials

by Prasolov, V. V.; Leites, Dimitry

• ISBN13: 9783540407140
• ISBN10: 3540407146
• Format: Hardcover
• Copyright: 2004-08-30
• Publisher: Springer Verlag

Summary

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. Many of the modern results have only been published in journals so far. 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. Galois theory is discussed primarily from the point of view of the theory of polynomials, not from that of the general theory of fields and their extensions.

Table of Contents

Foreword Notational conventions

Roots of polynomials

Inequalities for roots

The roots of a polynomial and of its derivative

The resultant and the discriminant

Separation of roots

Lagrange's series and estimates of the roots of a polynomial

Problems to

Solutions of selected problems

Irreducible polynomials

Main properties of irreducible polynomials

Irreducibility criteria

Irreducibility of trinomials and fournomials

Hilbert's irreducibility theorem

Algorithms for factorization into irreducible factors

Problems to

Solutions of selected problems

Polynomials of a particular form

Symmetric polynomials

Integer-valued polynomials

Cyclotomic polynomials

Chebyshev polynomials

Bernoulli's polynomials

Problems to

Solutions of selected problems

Certain properties of polynomials

Polynomials with prescribed values

The height of a polynomial and other norms

Equations for polynomials

Transformations of polynomials

Algebraic numbers

Problems to

Galois theory

Lagrange's theorem and the Galois resolvent

Basic Galois theory

How to solve equations by radicals

Calculations of the Galois groups

Ideals in polynomial rings

Hilbert's basis theorem and Hilbert's theorem on zeros

Grobner bases

Hilbert's seventeenth problem

The sums of squares: introduction

Artin's theory

Pfister's theory

Appendix

The Lenstra-Lenstra-Lovasz algorithm

Bibliography

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