Mathematical Lectures from Peking University For further volumes: www.springer.com/series/11574
Michel Broué Some Topics in Algebra An Advanced Undergraduate Course at PKU
Michel Broué Institut Universitaire de France Université Paris Diderot Paris 7 Paris, France ISSN 2197-4209 ISSN 2197-4217 (electronic) ISBN 978-3-642-41268-4 ISBN 978-3-642-41269-1 (ebook) DOI 10.1007/978-3-642-41269-1 Springer Heidelberg New York Dordrecht London Mathematics Subject Classification: 11-01, 12-01, 13-01 Springer-Verlag Berlin Heidelberg 2014 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface Never, before I had lectured in front of second and third years students of Peking University, did I feel that strongly how much mathematics are universal, a world where human minds think alike. And never before had I enjoyed it so much. The course I was supposed to give (an advanced undergraduate introduction to Algebra) had no specific syllabus. I decided to let it go, pushed or pulled by the student s reactions and my own feelings. It turned out to become an amazing and delightful encounter between, on one hand, ideas and discoveries of German mathematicians from the end of the XIXth and the beginning of the XXth centuries 1 revisited and taught by some French mathematicians from the XXth century 2, and, on the other hand, young brilliant Chinese students of the XXIst century. The pleasure of these students while discovering these concepts, results, examples, has been obvious all along the course, and even sometimes expressed loudly. Moreover, the speed of their understanding and handling notions which were mostly new to them was amazing. A couple of times, at the intermission, one of them came and politely told me that he thought he had found a more elegant proof than the one I had just given and each time he was indeed right, his proof was better, more elegant, more natural. Elegant, efficient, natural, pertinent, beautiful, clever, exciting: these are words sometimes heard when a mathematician discovers a new approach, a new proof, or even a new version of an old result. Whatever country, origin, culture that mathematician may be from: what is beautiful and pertinent for a German Herr Professor of the XIXth Century is also beautiful and pertinent for a young Chinese student of 2013. Of course, universality is not the peculiarity of mathematics, it is certainly shared by most of the arts, and partly by philosophy. But the essence of the universality of mathematics is not directly connected with feelings and events of any human life, pain or joy, love or disaster, war, freedom, death or future. Besides, the 1 Ideals were first defined by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (Lectures on Number Theory), after Ernst Kummer had introduced the concept of ideal numbers. The notion was later expanded by David Hilbert and Emmy Noether. 2 Like Nicolas Bourbaki. v
vi Preface universality of mathematics is a rule, almost a theorem: what is considered by all as good is indeed good. I do think this is one of the wonders of the world we live in. The more elegant proofs of the students are integrated in this book, without quotation to their authors since I did not know their names. This is one of the reasons why the book is dedicated to the students of PKU. CONVERSATION BETWEEN MATHEMATICIANS Anouk Grinberg
Preface vii Abstract During the Springs of 2011 and 2012, I was invited by the Beijing International Center for Mathematics Research to give an advanced undergraduate algebra course (once a week over two months each year). This is part of the Everest project of Chinese Education Ministry on first class students training. This book has been written during and for that course. By no way does it pretend to any type of exhaustivity. It is a quick and contingent introduction to Algebra in front of an extremely pleasant and passionate audience, heterogeneous but persistent. It certainly reflects some of my own tastes, and mainly the constraints of such a short period of teaching. A remark about the last two sections: following a well established tradition, we had planned to conclude by lecturing on the structure of finitely generated modules over principal ideal domains. But during the process of the course, after explaining that the notion of projective module is somehow more natural than the notion of free module, it became rather inevitable to replace principal ideal domains by Dedekind rings; this is less traditional in the literature but not really more difficult.
viii Preface Prerequisites This book requires a certain familiarity with the notions of groups, rings, fields, and specially with the undergraduate knowledge of linear algebra. More specifically, let k be a commutative field. We assume the reader knows the definition of the ring of polynomial k[x 1,...,X n ] in n indeterminates, the Euclidean division in Z and in k[x], as well as some of the consequences, like: both these rings are principal ideal domains, hence for p a prime number and P(X)an irreducible polynomial, both quotients Z/pZ and k[x]/(p (X)) are fields; the main results of an undergraduate course on k-linear algebra; matrices and their determinants. The following identity will not be proved: let M be an n n matrix with entries in k, let t Com(M) denote the transpose of its matrix of cofactors, let 1 n be the identity n n matrix; then t Com(M).M = det(m).1 n. We take for granted that the reader is familiar with the standard notation N (for numbers ) note that by convention N ={0, 1, 2,...}, Z for Zahlen ), Q (for quotients ), R (for reals ), C (for complexes ), as well as F p = Z/pZ (for finite ) see [7], p. 3. By convention, a field is a commutative ring where all nonzero elements are invertible. A noncommutative ring where all nonzero elements are invertible is called a division ring. We shall also use the following notation. For Ω any finite set, Ω will denote the number of its elements. For any sequence ξ 1,...,ξ n, or any product ξ 1 ξ n, and for all j = 1,...,n,we set (with obvious ad hoc convention for ξ 0 and ξ n+1 ): (ξ 1,..., ξ j,...,ξ n ) := (ξ 1,...,ξ j 1,ξ j+1,...,ξ n ), and ξ 1 ξ j ξ n := ξ 1 ξ j 1 ξ j+1 ξ n. A subset (subgroup, subring, submodule,...) Ω of a set (group, ring, module,...)ω is said to be proper if Ω Ω.
Acknowledgements I warmly thank Yanjun Liu, Xiong Huan, and specially Gunter Malle, for their careful and patient readings and corrections of the manuscript. My thanks go also to the students who attended the courses which motivated these notes, for their interest, their attention, their questions, their wonder for Mathematics. ix
Contents 1 Rings and Polynomial Algebras... 1 1.1 FirstDefinitions... 1 1.1.1 Rings... 1 1.1.2 Canonical Morphisms.... 9 1.1.3 Ideals... 12 1.1.4 Algebras... 15 1.1.5 Fields,DivisionRings... 20 1.2 Prime and Maximal Ideals, Integral Domains...... 36 1.2.1 Definition and First Examples... 36 1.2.2 Examples in Polynomial Rings... 37 1.2.3 NilradicalandRadical... 39 1.2.4 IntegralDomains,FieldsofFractions... 42 1.3 Divisibility and Irreducible Elements... 47 1.3.1 Divisors and Irreducible Elements... 47 1.3.2 Euclidean Rings... 49 1.3.3 GCDandLCM... 51 1.3.4 Case of Z[i] andapplication... 52 1.3.5 Irreducibility Criteria in R[X]... 54 1.4 Polynomial Rings in Several Indeterminates...... 66 1.4.1 Universal Property, Substitutions... 66 1.4.2 Transfer Properties...... 70 1.4.3 Symmetric Polynomials... 71 1.4.4 Resultant and Discriminant... 76 2 Modules... 87 2.1 DefinitionsandConventions... 87 2.1.1 Definitions... 87 2.1.2 Submodules... 90 2.1.3 Torsion Elements, Torsion Submodule..... 96 2.1.4 Free and Generating Systems, Free Modules... 98 2.1.5 Constructions: Direct Sums, Products, Tensor Products. 102 xi
xii Contents 2.1.6 Localization... 110 2.2 Finitely Generated Modules..... 115 2.2.1 Application: Integrality over a Ring... 116 2.2.2 Complement: Jacobson Rings, Hilbert s Nullstellensatz. 122 2.2.3 Noetherian Rings and Modules... 127 2.2.4 Finitely Generated Free Modules... 132 2.2.5 Finitely Generated Projective Modules..... 137 2.3 Finitely Generated Modules over Dedekind Domains... 146 2.3.1 Fractional Modules, Dedekind Domains.... 146 2.3.2 Finitely Generated Projective Modules over Dedekind Domains... 154 2.3.3 Remarkable Decompositions of Finitely Generated Torsion Modules... 159 2.3.4 Adapted Pseudo-bases for Submodules of Finitely Generated Projective Modules... 169 2.3.5 Applications to Abelian Groups and Endomorphisms of Vector Spaces... 175 2.4 Complement on Dedekind Domains... 186 2.4.1 Characterizations of Dedekind Domains.... 186 2.4.2 Rings of Integers of Number Fields Are Dedekind... 188 Erratum to: Rings and Polynomial Algebras... E1 References... 195 Index... 197