Algebra and Galois Theories
Galois theory has such close analogies with the theory of coverings that algebraists use a geometric language to speak of field extensions, while topologists speak of "Galois coverings". This book endeavors to develop these theories in a parallel way, sta
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Algebra and Galois Theories
Algebra and Galois Theories
M. C. Escher’s “Circle Limit III” © 2019 The M.C. Escher Company—The Netherlands. All rights reserved. www.mcescher.com
Régine Douady Adrien Douady •
Algebra and Galois Theories
123
Régine Douady Université Paris Denis-Diderot Paris, France
Adrien Douady (1935–2006) Université Paris-Sud Orsay Paris, France
Translated by Urmie Ray Sceaux, France
ISBN 978-3-030-32795-8 ISBN 978-3-030-32796-5 https://doi.org/10.1007/978-3-030-32796-5
(eBook)
Mathematics Subject Classification (2010): 15-01, 18-01, 55-01, 12F10, 14E20, 14F35, 14H, 14H30 Translation from the French language edition: Algèbre et théories galoisiennes by Régine and Adrien Douady, © Published by Cassini 2004. All Rights Reserved. © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, 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 physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Introduction
Similarities between Galois theory and the theory of covering spaces are so striking that algebraists use geometric language to talk of field extensions whereas topologists talk of Galois covers. Here we have tried to develop these theories in parallel, beginning with that of coverings. The reader will thereby be better able to visualize. This similarity can sometimes be found in specific formulations: (4.5.5) Proof of the proposition. Let u : . SðXÞ ! SðYÞ be a morphism in GThere is a morphism u : E SðXÞ ! E SðYÞ defined by u ðt; sÞ ¼ ðt; uðsÞÞ corresponding to the morphism u. The morphism u is compatible with the G-operation ? …
(5.7.4) Proof of the proposition. Let u : . SðBÞ ! SðAÞ be a morphism in GThere is a morphism of L-algebras u : LSðAÞ ! LSðBÞ defined by u ðhÞ ¼ h u corresponding to u. The homomorphism u is compatible with th
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