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.

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 DEGREESn 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 a

Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.

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.

Das vorliegende Blichlein ist aus Vorlesungen hervorgegangen, die wir abwechselnd an der Universitat Konstanz hielten und noch immer halten. Die Absicht dieser Vor- lesung ist es, Mathematikstudenten mittlerer Semester einen Einblick in die Mengen- lehre zu vermitteln, der ihnen gleichzeitig die flir die Mathematik wichtigsten mengen- theoretischen Begriffe und Satze an die Hand gibt. Diese Vorlesung halten wir gewohnlich zweistlindig im Sommersemester. Hieraus resultiert die Anzahl der Kapitel - jede Woche wird ein Kapitel besprochen. Wir setzen dabei eine gewisse Vertrautheit des Studenten im naiven Umgang mit Mengen aus den ersten Semestern voraus. Auch flihren wir bei Anwendungen der Mengenlehre nicht aile Beweise detailliert aus, sondern begnligen uns oft mit der Angabe der wich- tigsten Schritte. Dies gilt zum Beispiel flir den Autbau des Zahlsystems, speziell flir die Kapitel4 und 5. Urn in Kapitel 10 neben einfachen Anwendungen des Auswahl- axioms auch tieferliegende bringen zu konnen, sind wirt dort gezwungen, Vertraut- heit mit den Begriffen und Satzen der jeweiligen Theorie vorauszusetzen. Grundsatz- lich lassen sich jedoch aile in Beweisen bestehenden Lucken routinemall>ig schliell>en. Der von uns gewahlte Zugang zur Mengenlehre ist axiomatisch, vermeidet jedoch moglichst eine zu formale Darstellung. Wir versuchen, der mathematischen Praxis so nahe wie moglich zu bleiben, ohne dadurch allerdings eine mOgliche Formalisierbar- keit aus den Augen zu verlieren. Dber die Durchflihrung einer solchen Formalisierung (nach von Neumann, Godel, Bernays) berichten wir im Epilog.

Ein wesentliches Ziel dieses Buches ist, Studenten des Hauptstudiums und interessierten Mathematikern die Moeglichkeit zu eroeffnen, die bekanntesten, in der Algebra zur Zeit ublichen modelltheoretischen Schlusse kennen und verstehen zu lernen. Die Modelltheorie beschaftigt sich primar mit der Untersuchung der Modelle von Axiomensystemen, die in der Sprache der Logik erster Stufe formuliert sind. Die meisten, der in der Mathematik ublichen Axiomensystemen, gehoeren dazu.