Prasolov V.V. Polynomials

Springer, 2004. — 310 p. — (Algorithms and Computation in Mathematics 11). — ISBN 978-3-540-40714-0, 978-3-642-03979-9, 978-3-642-03980-5.The theory of polynomials constitutes an essential part of university courses of algebra and calculus. Nevertheless, there are very few books entirely devoted to this theory. Though, after the first Russian edition of this book was printed, there appeared several books devoted to particular aspects of the polynomial theory, they have almost no intersection with this book.
This book contains 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. 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. More precisely: In Chapter 1 we discuss, mostly classical, theorems about the distribution of the roots of a polynomial and of its derivative. It is also shown how to determine the number of real roots to a real polynomial, and how to separate them.
Chapter 2 deals with irreducibility criterions for polynomials with integer coefficients, and with algorithms for factorization of such polynomials and for polynomials with coefficients in the integers mod p.
In Chapter 3 we introduce and study some special classes of polynomials: symmetric (polynomials which are invariant when the indeterminates are permuted), integer valued (polynomials which attain integer values at all integer points), cyclotomic (polynomials with all primitive nth roots of unity as roots), and some interesting classes introduced by Chebyshev, and by Bernoulli. In Chapter 4 we collect a lot of scattered results on properties of polynomials. We discuss, e.g., how to construct polynomials with prescribed values in certain points (interpolation), how to represent a polynomial as a sum of powers of polynomials of degree one, and give a construction of numbers which are not roots of any polynomial with rational coefficients (transcendental numbers).
Chapter 5 is devoted to the classical Galois theory. It is well known that the roots of a polynomial equation of degree at most four in one variable can be expressed in terms of radicals of arithmetic expressions of its coefficients. A main application of Galois theory is that this is not possible in general for equations of degree five or higher.
In Chapter 6 three classical Hilbert’s theorems are given: an ideal in a polynomial ring has a finite basis (Hilbert’s basis theorem); if a polynomial f vanishes on all common zeros of f₁,…,f_r, then some power of f is a linear combination (with polynomial coefficients) of f₁,…,f_r (Hilbert’s Nullstellensatz); and if M = ⊕M_i is a finitely generated module over a polynomial ring over K, then dim_K Mi is a polynomial in i for large i (the Hilbert polynomial of M).
Furthermore, the theory of GrЁobner bases is introduced. GrЁobner bases are a tool for calculations in polynomial rings. An application is that solving systems of polynomial equations in several variables with finitely many solutions can be reduced to solving polynomial equations in one variable. In the final Chapter 7 considerable attention is given to Hilbert’s 17th problem on the representation of non-negative polynomials as the sum of squares of rational functions, and to its generalizations. The Lenstra-Lenstra- Lov`asz algorithm for factorization of polynomials with integer coefficients is discussed in an appendix.
Two important results of the theory of polynomials whose exposition requires quite a lot of space did not enter the book: how to solve fifth degree equations by means of theta functions, and the classification of commuting polynomials. These results are expounded in detail in two recently published books in which I directly participated: [Pr3] and [Pr4].Roots of Polynomials
Irreducible Polynomials
Polynomials of a Particular Form
Certain Properties of Polynomials
Galois Theory
Ideals in Polynomial Rings
Hilbert’s Seventeenth Problem
Appendix