This is an interesting, useful, well-organized, and well-written compendium of theorems and techniques about polynomials. It is especially strong on factorization, criteria for irreducibility, and separation of zeros. The present volume is a translation of the 2001 Russian second edition.
Other interesting topics covered include an exposition of Galois theory aimed the classical problem of solvability by radicals, but also showing methods for finding the Galois group of particular polynomials. These methods include inclusion criteria for the Galois group, the relation between Galois groups over the rationals and over the integers mod p, and a complete determination of the group for any cubic or quartic polynomial. Most of the book deals with general properties of polynomials in a single variable. Functions of several variables are also covered, although more theoretically, in a chapter on ideals in polynomial rings, covering Hilbert’s basis theorem, Hilbert’s Nullstellensatz, and Gröbner bases. There is a chapter on particular types of polynomials, with many properties derived for cyclotomic, Chebyshev, and Legendre polynomials.
This is primarily a reference work, not a textbook, although it does include a set of interesting problems (with some solutions) at the end if each chapter. Barbeau’s Polynomials is a good polynomial book, that is intended as a text and has a logical progression through the subject. It is aimed much lower and is much less comprehensive that the present book, but still has much useful information.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.