Algebra

Undergraduate Texts in Mathematics
Editors

S. Axler F.W. Gehring K.A. Ribet

Antoine Chambert-Loir

A Field Guide to Algebra
With 12 Illustrations

Antoine Chambert-Loir Universite de Rennes 1 ´ IRMAR, Campus de Beaulieu 35042 Rennes Cedex France

Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W. Gehring MathematicsDepartment East Hall University of Michigan Ann Arbor, MI 48109 USA K.A. Ribet Department of Mathematics University of California at Berkeley Berkeley, CA 94720-3840 USA

Pictured on the cover: Constructing the square root of a real number (Pythagoras’s theorem). See page 2 for discussion.

Mathematics Subject Classification (2000): 12-01, 12Fxx, 11Rxx, 13Bxx Library of CongressCataloging-in-Publication Data Chambert-Loir, Antoine. A field guide to algebra / Antoine Chambert-Loir. p. cm. — (Undergraduate texts in mathematics, ISSN 0172-6056) Includes bibliographical references and index. ISBN 0-387-21428-3 (hardback : alk. paper) 1. Algebraic fields. I. Title. II. Series. QA247.C48 2004 512′.3—dc22 2004048103 ISBN 0-387-21428-3 Printed on acid-free paper.

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Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavour to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; nor again, like merchants or retail-traders, with a view tobuying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being. Plato, Republic, Book VII. Engl. transl. by B. Jowett J’ai fait en analyse plusieurs choses nouvelles. ´ Evariste Galois, letter to A. Chevalier (29 mai 1832)

Contents

1

Field extensions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Constructions with ruler and compass . . . . . . . . . . . . . . . . . . . . . . 1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Some classical impossibilities . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Appendix: Transcendence of e and π . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roots . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Ring of remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Splitting extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Algebraically closed fields; algebraic closure . . . . . . . . . . . . . . . . . 2.4 Appendix: Structure of polynomial rings . . . . ....