Real Algebra

A First Course

Manfred Knebusch, Claus Scheiderer (Autoren)

Buch | Softcover

XII, 206 Seiten

2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-09799-7 (ISBN)

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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 in linear algebra, field theory and ring theory. It also provides a carefully written reference for specialists in real algebra, real algebraic geometry and related fields.

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.

1 Ordered fields and their real closures.- 2 Convex valuation rings and real places.- 3 The real spectrum.- 4 Recent developments.

"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." (Tim Netzer, zbMATH 1505.13001, 2023)

“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.” (Tim Netzer, zbMATH 1505.13001, 2023)

Erscheinungsdatum	25.10.2022
Reihe/Serie	Universitext
Co-Autor	Thomas Unger
Übersetzer	Thomas Unger
Zusatzinfo	XII, 206 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	340 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	hilbert's 17th problem • Nullstellensatz • ordered fields • ordered rings • orderings • Positivstellensatz • preorderings • Real Algebra • real algebraic geometry • real spectrum • semialgebraic sets • spectral space • valuation rings • valuations
ISBN-10	3-031-09799-8 / 3031097998
ISBN-13	978-3-031-09799-7 / 9783031097997
Zustand	Neuware