Formally p-adic Fields
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1050
Alexander Prestel Peter Raquette
Formally p-adic Fields
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Authors Alexander Prestel Fakultat fur Mathematik, Universitat Konstanz Postfach 5560, 7750 Konstanz, Federal Republic of Germany Peter Roquette Mathematisches Institut, Universitat Heidelberg 1m Neuenheimer Feld 288, 6900 Heidelberg, Federal Republic of Germany
This book is also available as no. 38 of the series "Monografias de Maternatica", published by the Instituto de Matematica Pura e Aplicada, Rio de Janeiro
AMS Subject Classifications (1980): 12B99, 12JlO ISBN 3-540-12890-5 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12890-5 SpringerVeriag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Preface These notes result from lectures given by the authors at IMPA in Rio de Janeiro - in 1980 by the second and in 1982 by the first author. The 1980 course was mainly concerned with the content of Sections 6 and 7 , using as a prerequesite the Ax-KochenErsov- Theorem
on the model completeness of the theory of
p-adically closed fields. After that, an algebraic approach to this important theorem
as well as an analysis of p-adic closures
was developed (contained in Sections 3 and 4). The present lecture notes essentially coincide with the 1982 course. In the introductory Section 1 we try to point out the analogy between the theory of p-valued fields and the well-known theory of ordered fields. After giving some basic definitions and examples as well as basic facts from general valuation theory in Section 2, we develop the theory of p-valued fields, i.e. fields together with a fixed p-valuation in Section 3 and 4 . From Section 6 on we no longer fix a certain p-valuation, instead we only assume the existence of such a valuation. Fields which admit some p-valuation are called formally p-adic. The theory of formally p-adic fields is concerned with the investigation of all pvaluations rather than just one. In Section 7 we concentrate on the important case of function fields. In Section 5 we use results proved in previous sections for model theoretic investigations of formally p-adic fields. In particular we deduce the Ax-KochenErsov-Theorem. Viewed historically, these results stand at the beginning of the development of formally p-adic fields.
IV
The only Section which makes use of model theoretic notions and facts is Section 5 . However, there is one exception - the notion of saturated structures - which is used also in the formulation
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