About the Book
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.
This comprehensive book covers both long-standing results in the theory of polynomials and recent developments which have until now only been available in the research literature. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials, including those which are symmetric, integer-value or cyclotomic, and those of Chebyshev and Bernoulli. There follow chapters on Galois theory and ideals in polynomial rings. Finally there is a detailed discussion of Hilbert's 17th problem on the representation of non-negative polynomials as sums of squares of rational functions and generalizations.
From the reviews:
..". Despite the appearance of this book in a series titled Algorithms and Computation of Mathematics, computation occupies only a small part of the monograph. It is best described as a useful reference for one's personal collection and a text for a full-year course given to graduate or even senior undergraduate students. .....] the book under review is worth purchasing for the library and possibly even for one's own collection. The author's interest in the history and development of this area is evident, and we have pleasant glimpses of progress over the last three centuries. He exercises nice judgment in selection of arguments, with respect to both representativeness of approaches and elegance, so that the reader gains a synopsis of and guide to the literature, in which more detail can be found. ..." (E. Barbeau, SIAM Review 47, No. 3, 2005)
..". the volume is packed with results and proofs that are well organised thematically into chapters and sections. What is unusual is to have a text that embraces and intermingles both analytic and algebraic aspects of the theory. Although the subject is about such basic objects, many tough results of considerable generality are incorporated and it is striking that refinements, both in theorems and proofs continued throughout the latter part of the Twentieth Century. ...] There is a plentiful of problems, some of which might be challenging even for polynomial people; solutions to selected problems are also included." (S.D.Cohen, MathSciNet, MR 2082772, 2005)
"Problems concerning polynomials have impulsed resp. accompanied the development of algebra from its very beginning until today and over the centuries a lot of mathematical gems have been brought to light. This book presents a few of them, some being classical, but partly probably unknown even to experts, some being quite recently discovered. ...] Many historical comments and a clear style make the book very readable, so it can be recommended warmly to non-experts already at an undergraduate level and, because of its contents, to experts as well." (G.Kowol, Monatshefte fur Mathematik 146, Issue 4, 2005)