Bøger af Manfred Knebusch

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.

The specialization theory of quadratic and symmetric bilinear forms over fields and the subsequent generic splitting theory of quadratic forms were invented by the author in the mid-1970's. They came to fruition in the ensuing decades and have become an integral part of the geometric methods in quadratic form theory. This book comprehensively covers the specialization and generic splitting theories. These theories, originally developed mainly for fields of characteristic different from 2, are explored here without this restriction. In this book, a quadratic form ¿ over a field of characteristic 2 is allowed to have a big quasilinear part QL(¿) (defined as the restriction of ¿ to the radical of the bilinear form associated to ¿), while in most of the literature QL(¿) is assumed to have dimension at most 1. Of course, in nature, quadratic forms with a big quasilinear part abound.In addition to chapters on specialization theory, generic splitting theory and their applications, the book's finalchapter contains research never before published on specialization with respect to quadratic places and will provide the reader with a glimpse towards the future.

The Prufer extensions of a commutative ring A are roughly those commutative ring extensions R / A, where commutative algebra is governed by Manis valuations on R with integral values on A. While in Volume I Prufer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations.