Valuation Theory

These are the revised notes of a course for graduate students and some seminar talks which I gave at the University of Rochester during Fall Term 1969/70. They would not have been written without the encouragement and the aid which I received, during all stages of the work, by friends from Rochester, Rio de Janeiro, and Bonn. I wish to thank all of them: Barbara Grabkowicz encouraged me to write these notes in English and read carefully parts of a preliminary manuscript, as did Gervasio G. Bastos, Yves A. E. Lequain, Walter Strubel, and Antonio J. Engler. Many valuable suggestions were given me by Yves A. E. Lequain, and several improvements of theorems and proofs are due to him. I am particularly grateful to Linda C. Hill for her criticism in reading the last version and for improving and smoothing many of my formulations. Last but not least I thank Wilson Goes for the excellent typing. Most of this book was elaborated when I stayed in Rio de Janeiro as a Visiting Professor at IMPA (Institute for Pure and Applied Mathematics) and as a Pesquisador-Conferencista of CNPq (National Research Council). Thanks are also due to these institu tions.

Inhalt
I Valuations.- § 1 Valuations.- § 2 Completions and extensions of valuations.- § 3 Non-archimedean valuations.- § 4 Discrete exponential valuations.- § 5 Complete discretely valued fields.- II Valuation Rings.- § 6 Valuation rings.- § 7 Krull valuations.- § 8 Places.- § 9 The extension theorem.- § 10 Integrally closed domains 66.- § 11 Prüfer rings. Approximation theorems.- § 12 Krull rings and Dedekind rings.- III Extension of Valuation Rings.- § 13 The case of an algebraic field extension.- § 14 The case of a normal field extension.- § 15 Decomposition group and decomposition field.- § 16 Henselian valuation rings.- § 17 Extension of valuation rings and henselization.- § 18 The equality ? ei · fi = n.- § 19 Inertia group and inertia field.- § 20 Ramification group and ramification field.- § 21 Higher ramification groups.- § 22 Unramified and tamely ramified extensions.- IV Fields with Prescribed Valuations.- § 23 Introduction and notation.- § 24 Topological preliminaries.- § 25 Solvable (?1,...,?k)-prescriptions.- § 26 Henselian and antihenselian valuations.- § 27 Prescription of value groups and residue fields.- § 28 The case of infinite field extensions.- Exercises.