Vector Spaces and Field Extensions
Up to now, we have kept our attention focused on the field ℚ and its p-adic completions. We have already felt, however, the need to consider other fields (for example, when we dealt with the zeros of a function defined by a power series). In fact, just as
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Fernando Q. Gouvea
p-adic Numbers An Introduction With 15 Figures
Springer-Verlag Berlin Heidelberg GmbH
Fernando Q. Gouvea Department of Mathematics and Computer Science Colby College Waterville, ME 04901, USA
Mathematics Subject Classification (1991): 11-0 I, lIS-xx
ISBN 978-3-540-56844-5
Library of Congress Cataloging. in-Publication Data Gouvea, Fernando, 1957p-adic numbers: an introduction/Fernando Gouvea. p. cm. - (Universitext) ISBN 978-3-540-56844-5 ISBN 978-3-662-22278-2 (eBook) DOI 10.1007/978-3-662-22278-2 I. p-adic numbers. I. Title. II. Series. QA24l.G64 1993 512'.74-dc20 93-25593 CIP 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 microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag in 1993
41/3140 - 5 4 3 2 10 - Printed on acid-free paper
Contents
Introduction 1
Aperitif . 1.1 Hensel's Analogy 1.2 Solving Congruences Modulo pn . 1.3 Other Examples
2 Foundations. . . . . 2.1 Absolute Values on a Field 2.2 Basic Properties 2.3 Topology 2.4 Algebra . . .
1 5
5 12
17 21 21
27 29 37
3 p-adic Numbers 3.1 Absolute Values on Q 3.2 Completions . . . 3.3 Exploring Qp .. 3.4 Hensel's Lemma 3.5 Local and Global
41 41
4 Elementary Analysis in Qp 4.1 Sequences and Series .. . 4.2 Power Series . . . . . . . . 4.3 Some Elementary Functions 4.4 Interpolation .. . . . . . .
85
5 Vector Spaces and Field Extensions 5.1 Normed Vector Spaces over Complete Valued Fields 5.2 Finite-dimensional Normed Vector Spaces 5.3 Finite Field Extensions . . . . . 5.4 Properties of Finite Extensions . . . . . . 5.5 Analysis . . . . . . . . . . . . . . . . . . . 5.6 Example: Adjoining a p-th Root of Unity 5.7 On to
0 there exists an n such that the partial
sum
is divisible by 2M. Problem 23 Can you give a direct proof of that fact?
What this example points out is that using p-adic methods, and in particular the methods of the calculus in the p-adic context, we can often prove facts about divisibility by powers of p which are otherwise quite hard to understand. The proofs are often, as in this case, "cleaner" than any direct proof would be, and therefore easier to understand. We will look at many more examples of this before we are done.
2
Foundations
The goal of this chapter is to begin to lay a solid foundation for the theory we described informally in Chapter One. The main idea will be to introduce a different absolute value function on the field of rational numbers. This will give us a different way to measure distances, hence a different c
