An introduction to real algebra
T. Y. Lam · Rocky Mountain Journal of Mathematics · 1984
systematic study of real algebra over rings, as well as its applications to real algebraic geometry.In these notes, we shall give a self-contained introduction to these recent developments in real algebra.We shall not try to go deeply into any single topic, nor shall we try to formulate results in their greatest possible degree of generality.Instead, we shall focus on what we perceive to be the central facts, and try to organize them, with some care, into a coherent picture.By aiming our exposition at the nonexperts, it is hoped that this work will make the techniques of real algebra accessible to a general audience.For the experts, my exposition has probably little to offer, except perhaps by way of terminology.Since the subject of real algebra over rings is relatively new, the terminology used so far in the literature seemed to be in a great disarray.Needless to say, the lack of a consistent terminology in a subject is not in the best interest of its development.In view of this, I take the opportunity here to offer a remedy.In these notes, we shall call a ring A semireal if -1 is not a sum of squares in A\ we shall call A real (or formally real) if £ 0? = 0 =*> all Û,= 0 in A* If (A, STO) is a local ring, we shall call A residually real if the residue fieli A/Wl is real.By an ordering on a ring A, we mean (essentially) an ordering on the quotient field of A/p for some prime ideal p cz A. For us, thfe system of terminology works very well throughout these notes.For better or for worse, we propose it for possible use by the mathematical community.Since this work is intended to be an exposition and not a broad survey» we have left out the discussion of many important topics.To get a broader view of the subject of real algebra, we urge our readers to consult also the other available survey/expository articles on the subject, for instant [41, [18] and [19].Readers desiring more information about the recerf literature should consult the two excellent collections of articles [161 &ß [25] listed in the references.The work presented here is a revised and somewhat expanded version of my lecture notes circulated at the Sexta Escuela Latino Americana * Matematicas in Oaxtepec, Mexico, July, 1982.I thank Professor J<#* Adem, organizer of ELAM VI, for his kind permission for me to use tfc* same material for the Real Algebraic Geometry Conference.During d* preparation of these notes, I have had the great benefit of frequent co*" sultation with J. Merzel and A. Prestel.Both of them have generou^ given their time in helping me out when I got stuck in the writing.1&* valuable suggestions have resulted in many substantial improvements 0