Algebraic Function Fields of One Variable
Sections 3.1–3.4 survey the theory of functions of one variable; the Riemann-Roch Theorem; properties of holomorphy rings of function fields; and extensions of the field of constants.
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A Series of Modern Surveys in Mathematics
Editorial Board M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton H. W. Lenstra, Jr., Leiden J. Tits, Paris D. B. Zagier, Bonn G. Ziegler, Berlin Managing Editor R. Remmert, Münster
Volume 11
Michael D. Fried • Moshe Jarden
Field Arithmetic Third Edition Revised by Moshe Jarden
Michael D. Fried Department of Mathematics Montana State University – Billings Billings MT 59101 USA [email protected]
ISBN 978-3-540-77269-9
Moshe Jarden School of Mathematics Tel Aviv University Ramat Aviv, Tel Aviv 69978 Israel [email protected]
e-ISBN 978-3-540-77270-5
DOI 10.1007/978-3-540-77270-5 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISSN 0071-1136 Library of Congress Control Number: 2008924174 Mathematics Subject Classification (2000): 12E30 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
To those precious colleagues who can appreciate the goals of and connections to other areas. To those who acknowledge the depth of what we already know from the absorbed contribution of previous generations before we address our papers. To those who can transcend the hubris of today’s mathematical community.
Table of Contents Chapter 1. Infinite Galois Theory and Profinite Groups 1.1 Inverse Limits . . . . . . . . . . . . . . . 1.2 Profinite Groups . . . . . . . . . . . . . . 1.3 Infinite Galois Theory . . . . . . . . . . . . 1.4 The p-adic Integers and the Pr¨ ufer Group . . . 1.5 The Absolute Galois Group of a Finite Field . . Exercises . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . .
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Chapter 2. Valuations and Linear Disjointness . 2.1 Valuations, Places, and Valuation Rings . . 2.2 Discrete Valuations . . . . . . . . . . . 2.3 Extensions of Valuations and Places. . . . 2.4 Integral Extensions and Dedekind Domains 2.5 Linear Disjointness of Fields . . . . . . . 2.6 Separable, Regular, and Primary Extensions 2.7 The Imperfect Degree of a Field . . . . . 2.8
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