This book provides an introduction to fundamental methods and techniques of algebra over ordered fields. It is a revised and updated translation of the classic textbook Einführung in die reelle Algebra. Beginning with the basics of ordered fields and their real closures, the book proceeds to discuss methods for counting the number of real roots of polynomials. Followed by a thorough introduction to Krull valuations, this culminates in Artin's solution of Hilbert's 17th Problem. Next, the fundamental concept of the real spectrum of a commutative ring is introduced with applications. The final chapter gives a brief overview of important developments in real algebra and geometry-as far as they are directly related to the contents of the earlier chapters-since the publication of the original German edition. Real Algebra is aimed at advanced undergraduate and beginning graduate students who have a good grounding inlinear algebra, field theory and ring theory. It also provides a carefully written reference for specialists in real algebra, real algebraic geometry and related fields.

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Manfred Knebusch is Professor Emeritus at the University of Regensburg. He has written nine books and more than 80 papers on the algebraic theory of quadratic forms over rings and fields, valuation theory, real algebra and real algebraic geometry. His current research focusses on tropical geometry. Claus Scheiderer is Professor at Konstanz University. His primary research interests are real algebraic geometry and convex algebraic geometry. Thomas Unger is Associate Professor at University College Dublin. His research interests include quadratic and hermitian forms, algebras with involution, and noncommutative real algebra and geometry.