Alexander Prestel Livres

Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -for instance to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values alone.

Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.

Inhaltsverzeichnis1 Logik 1. Stufe.1.1 Analyse mathematischer Beweise.1.2 Aufbau formaler Sprachen.1.3 Formale Beweise.1.4 Vollständigkeit der Logik 1. Stufe.1.5 Semantik 1. Stufe.1.6 Axiomatisierung einiger mathematischer Theorien.Übungen zu Kapitel 1.2 Modellkonstruktionen.2.1 Termmodelle.2.2 Morphismen von Strukturen.2.3 Substrukturen.2.4 Elementare Erweiterungen und Ketten.2.5 Saturierte Strukturen.2.6 Ultraprodukte.Übungen zu Kapitel 2.3 Eigenschaften von Modellklassen.3.1 Kompaktheit und Separation.3.2 Kategorizität.3.3 Modellvollständigkeit.3.4 Quantorenelimination.Übungen zu Kapitel 3.4 Modelltheorie einiger algebraischer Theorien.4.1 Angeordnete abelsche Gruppen.4.2 Angeordnete Körper.4.3 Bewertete Körper: Beispiele und Eigenschaften.4.4 Algebraisch abgeschlossene bewertete Körper.4.5 Reell abgeschlossene bewertete Körper.4.6 Henselsche Körper.Übungen zu Kapitel 4.Anhang. Bemerkungen zur Entscheidbarkeit.Literaturhinweise.Symbolverzeichnis.Namen- und Sachwortverzeichnis.