Lectures on formally real fields / Alexander Prestel

This is a reprint with only a few minor changes of the author's book of the same title published in 1975 by IMPA of Brazil. It is still an excellent introduction to the theory of formally real fields (i.e., ordered fields), including the connections with model theory, quadratic form theory, and valuation theory. The new edition makes the book available to a much wider audience; this is especially fortunate since it has been referred to in many research papers. For information on the substantial developments in ordered fields and related areas since 1975, one may consult the book by T.-Y. Lam [Orderings, valuations and quadratic forms, Amer. Math. Soc., Providence, R.I., 1983; MR0714331 (85e:11024)] and the survey article by M. Knebusch [Quadratic and Hermitian forms (Hamilton, Ont., 1983), 51–105, Amer. Math. Soc., Providence, R.I., 1984].
A distinctive feature of this book is its use of model theory to obtain algebraic results. There is a good brief introduction to model theory which is self-contained except for omitting the proof of the compactness theorem. This is followed by proofs of Tarski's theorem on elimination of quantifiers for real-closed fields and Tarski's principle of elementary equivalence of real-closed fields. The power of these theorems is nicely illustrated by concise proofs of Artin's solution to Hilbert's 17th problem and of the Dubois-Risler real Nullstellensatz for real-closed fields. In all, this takes up about one-quarter of the book, while the rest is independent of the model theory. (MathSciNet)