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Unformatted text preview: NICOLAS BOURBAKI Elements of Mathematics Algebra I Chapters 13 HERMANN, PUBLISHERS IN ARTS AND SCIENCE 293 rue Lecourbe, 75015 Paris, France A vv ADDISONWESLEY PUBLISHING COMPANY Advanced Book Program Reading, Massachusetts Originally published as ELEMENTS DE MATHEMATIQUE, ALGEBRE 0 1943, 1947, 1948, 1971 by Hermann, Paris ISBN 2705656758 (Hermann) ISBN 0201006391 (AddisonWesley) Library of Congress catalog card number LC 725558 American Mathematical Society (MOS) Subject Classification Scheme (1970) : 15A03, 15A69, 15A75, 15A78 Printed in Great Britain 0 1974 by Hermann. All rights reserved This book, or parts thereof, may not be reproduced in any form without the publisher’s written permission TO THE READER 1. This series of volumes, a list of which is given on pages ix and x, takes up mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledgc of mathematics on the rcaders’ part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course. 2. The method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular. This choice has been dictated by the main purpose of the treatise, which is to provide a solid foundation for the whole body of modern mathematics. For this it is indis pensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands of proof impose a rigorously fixed order on the subject matter. It follows that the utility of certain considerations will not be immediately apparent to the reader unless he has already a fairly extended knowledge of mathematics; otherwise he must have the patience to suspend judgment until the occasion arises. 3. In order to mitigate this disadvantage we have frequently inserted examples in the text which refer to facts the reader may already know but which have not yet been discussed in the series. Such examples are always placed between two asterisks: * . . *. Most readers will undoubtedly find that these examples will help them to understand the text, and will prefer not to leave them out, even at a first reading. Their omission would of course have no disadvantage, from a purely logical point of view. 4. This series is divided into volumes (here called “Books”). The first six Books are numbered and, in general, every statement in the text assumes as known only those results which have already been discussed in the preceding V TOTHEREADER TO THE READER volumes. This rule holds good within each Book, but for convenience of expo sition these Books are no longer arranged in a consecutive order. At the begin ning of each of these Books (or of these chapters), the reader will find a precise indication of its logical relationship to the other Books and he will thus be...
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