Real algebra. A first course. Translated from the German and with contributions by Thomas Unger (Q2160152)

The present textbook is a translation of the 1989 German book [\textit{M. Knebusch} and \textit{C. Scheiderer}, Einführung in die reelle Algebra. (Introduction to real algebra). Braunschweig etc.: Friredr. Vieweg \&| Sohn (1989; Zbl 0732.12001)], with some gentle corrections and a short additional chapter on recent developments. The book provides an elementary introduction to concepts and results from real algebra. Real algebra is the algebraic theory behind semialgebraic geometry, just as commutative algebra is the algebraic theory behind algebraic geometry. The theory in the book is elementary in the sense that is does not require much knowledge beyond linear algebra and basic theory of groups, rings and fields. The first chapter develops the theory of real closed fields, in combination with quadratic form theory. I also contains a nice and thorough account of methods to count real zeros of polynomials, dating back to the 19th century (Sturm, Hermite, Sylvester \dots) The second chapter contains an introduction to the theory of valuations and their interplay with field orderings. It culminates in a proof of Hilbert's 17th problem, without using model theoretic concepts. The third chapter is devoted to the real spectrum, a ``real space'' associated to a commutative ring, in which abstract semialgebraic geometry can be performed. It contains several important Positiv- and Nullstellensätze, and explains connections to valuation theory, among others. Chapter four is new, containing a very brief overview of important developments since the publication of the original version of the book. These include the theory of fewnomials, more on quadratic forms, results on the number of inequalities needed to define a semialgebraic set, and denominator-free, as well as non-commutative Stellensätze. More than 30 years after its initial publication, the present textbook is still a very valuable source for results in real algebra. It can serve as a textbook for a university course, but also experts will benefit from the nice account of concepts and results. It's great that the book is available again, in particular in an English translation for an international audience.