P.M. Cohn. Basic Algebra. Groups, Rings and Fields. m Springer

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1 Basic Algebra

2 P.M. Cohn Basic Algebra Groups, Rings and Fields m Springer

3 P.M. Cohn, MA, PhD, FRS Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK British Library Cataloguing in Publication Data Cohn, P. M. (Paul Moritz) Basic algebra: groups, rings and fields I. Algebra 2. Rings (Algebra) 3. Algebraic fields I. Title 512 ISBN Library of Congress Cataloging-in-Publication Data Cohn, P.M. (Paul Moritz) Basic algebra: groups, rings, and fields/p.m. Cohn, p. cm. Includes bibliographical references and indexes. ISBN ISBN (ebook) DOI / Algebra. I. Title. QA154.3.C dc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. springeronline.com Professor P.M. Cohn 2003 Originally published by Springer-Verlag London Berlin Heidelberg in 2003 Softcover reprint of the hardcover 1st edition nd printing 2005 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting by BC Typesetting, Bristol BS31 1NZ 12/ Printed on acid-free paper SPIN

4 Contents Preface... Conventions on Terminology... ix xi 1. Sets Finite, Countable and Uncountable Sets.... Zorn's Lemma and Well-ordered Sets.... Graphs Groups 2.1 Definition and Basic Properties Permutation Groups The Isomorphism Theorems Soluble and Nilpotent Groups Commutators The Frattini Subgroup and the Fitting Subgroup Lattices and Categories 3.1 Definitions; Modular and Distributive Lattices Chain Conditions Categories Boolean Algebras Rings and Modules 4.1 The Definitions Recalled The Category of Modules over a Ring Semisimple Modules Matrix Rings Direct Products of Rings Free Modules Projective and Injective Modules The Tensor Product of Modules Duality of Finite Abelian Groups v

5 vi Basic Algebra 5. Algebras 5.1 Algebras; Definition and Examples The Wedderburn Structure Theorems The Radical The Tensor Product of Algebras The Regular Representation; Norm and Trace Mobius Functions Multilinear Algebra 6.1 Graded Algebras Free Algebras and Tensor Algebras The Hilbert Series of a Graded Ring or Module The Exterior Algebra on a Module Field Theory 7.1 Fields and their Extensions Splitting Fields The Algebraic Closure of a Field Separability Automorphisms of Field Extensions The Fundamental Theorem of Galois Theory Roots of Unity Finite Fields Primitive Elements; Norm and Trace Galois Theory of Equations The Solution of Equations by Radicals Quadratic Forms and Ordered Fields 8.1 Inner Product Spaces Orthogonal Sums and Diagonalization The Orthogonal Group of a Space The Clifford Algebra and the Spinor Norm Witt's Cancellation Theorem and the Witt Group of a Field Ordered Fields The Field of Real Numbers Formally Real Fields The Witt Ring of a Field The Symplectic Group Quadratic Forms in Characteristic Two Valuation Theory 9.1 Divisibility and Valuations Absolute Values The p-adic Numbers Integral Elements Extension of Valuations

6 Contents vii 10. Commutative Rings 10.1 Operations on Ideals Prime Ideals and Factorization Localization loa Noetherian Rings Dedekind Domains Modules over Dedekind Domains Algebraic Equations The Primary Decomposition Dimension The Hilbert Nullstellensatz Infinite Field Extensions 11.1 Abstract Dependence Relations Algebraic Dependence Simple Transcendental Extensions Separable and p-radical Extensions Derivations Linearly Disjoint Extensions Composites of Fields Infinite Algebraic Extensions Galois Descent Kummer Extensions Bibliography List of Notations Author Index Subject Index

7 Preface Much of the second and third year undergraduate course in mathematics (as well as some graduate work) was covered by Volumes 2 and 3 of my book on algebra, now out of print. 1 So I was very pleased when Springer Verlag offered to bring out a new version of these volumes. The present book is based on both these volumes, complemented by the definitions and basic facts on groups and rings. Thus the volume is addressed to students who have some knowledge of linear algebra and who have met groups and fields, though all the essential facts are recalled here. My overall aim has been to present as many of the important results in algebra as would conveniently fit into one volume. It is my hope to collect the remaining parts of Volumes 2 and 3 into a second book, more oriented towards applications. 2 Apart from chapters on groups (Chapter 2), rings and modules (Chapters 4, 5 and 6) and fields (Chapters 7 and 11), a number of concepts are treated that are less central but nevertheless have many uses. Chapter 1, on set theory, deals with countable and well-ordered sets, as well as Zorn's lemma and a brief section on graphs. Chapter 3 introduces lattices and categories, both concepts that form an important part of the language of modern algebra. The general theory of quadratic forms has many links with ordered fields, which are developed in Chapter 8. Chapters 9 and 10 are devoted to valuation theory and commutative rings, a subject that has gained in importance through its use in algebraic geometry. On a first encounter some readers may find the style of this book somewhat concise, but they should bear in mind that mathematical texts are best read with paper and pencil, to work out the full consequences of what is being said and to check examples. The matter has been well put by Einstein, who said: "Everything should be explained as far as possible but no further." There are numerous exercises throughout, with occasional hints (but no solutions), and some historical remarks. My thanks are due to the staff of Springer Verlag for the efficient way they have produced this volume. University College London P.M. Cohn June 2002 I Algebra, Vol. 2 (2nd edn, 1989) and Vol. 3 (2nd edn, 1991), Wiley and Sons. 2 Further Algebra and Applications, Springer Verlag, London (2003). Referred to in the text as FA. ix

8 Conventions on Terminology We assume that our readers are acquainted with the notion of a set (and even with groups and rings, though their definitions will be recalled in Chapters 2, 4). They will have seen notations such as x E 5 (x is a member of 5), 5' S; 5 or 5 ;2 5' (5' is a subset of 5) and T C 5 or 5 :) T (T is a proper subset of 5) and 0 for the empty set. For any propositions P, Q we write 'p::::} Q' or 'Q {::: P' to indicate that P implies Q, and 'p # Q' to mean 'p::::} Q and Q::::} P', i.e. that P is equivalent to Q. A property (of members of a set 5) is said to hold for almost all members of 5 if it holds for all but a finite number of members of 5. If T is a subset of 5, its complement in 5 will be denoted by 5\ T. This notation is also used occasionally for the left coset space (see Section 2.1); the risk of confusion is small. We can list the elements of a set 5 by indexing them, e.g. if 5 is finite, with n elements, we can write 5 = {Xl, X2,..., xn }; we also write 151 = n. More generally, any set can be indexed by a suitable indexing set: 5 = {X).hE!' where I is the indexing set. A set in this form is often called a family indexed by I; it is in effect prescribing a mapping from I to 5. This mapping is generally not assumed to be injective, thus x). may equal xfj- even if A =/: M. All mappings between sets are as a rule written on the right, so that fg means: first f, then g. Iff: T, i.e. f is a mapping from 5 to T and 5' is a subset of 5, then the restriction of f to 5' is denoted by fi5'. A mapping f : T is called injective or one-one if different members of 5 have different images, surjective or onto if every member of T is an image of some member of 5, and bijective if it is both injective and surjective. Mappings are often arranged as diagrams (see Section 4.2); a diagram is commutative if the different ways of going from one point to another along the arrows give the same result. Frequently a two-index expression f(i, j) is equal to 1 if i = j and 0 otherwise. This is indicated by using the Kronecker symbol 8 ij; thus f(i, j) = 8 ij A set 5 is partially ordered, often just called ordered, if there is a binary relation :s, called a partial ordering, defined on 5 with the properties: 0.1 x:s X for all X E 5 (reflexive), 0.2 X :s y, y :s z ::::} x :s z for all x, y, z E 5 (transitive), 0.3 x :s y, y :s x ::::} x = y for all x, y E 5 (antisymmetric). If only 0.1 and 0.2 hold, we speak of a preordering. The ordering is total if any two elements are comparable, i.e. x :s y or y :s x for any x, y E 5. If ':s' is a partial ordering on a set 5, we shall write 'x < y' (x is strictly xi

9 xii Basic Algebra less than y) to mean 'x.::s y and x =1= y', and we write x :::: y, x > y for y.::s x, y < x respectively. As is easily verified, the opposite ordering '::::' again satisfies and so is again a partial ordering. Thus any general statement about ordered sets has a dual, which is obtained by interpreting the original statement for the oppositely ordered set. This principle can often be used to shorten proofs. A binary relation ~ on a set 5 is called an equivalence relation if it is reflexive, transitive and symmetric, i.e. x y ~ => y ~ x for, all x, y E 5. For example, equality is an equivalence relation. Given an equivalence on 5, we can list all members equivalent to a given one together in a class, and in this way obtain a partition of 5 into a collection of disjoint subsets, the equivalence classes or blocks. The set of equivalence classes is denoted by 51 ~ and is called the quotient set of 5 by the equivalence ~. Given sets 5, T, their Cartesian or direct product, denoted by 5 x T, is the set of pairs (x,y), where x E 5, YET. If 5, T are any ordered sets, their direct product can again be ordered by writing (x, y).::s (x', y') to mean: x.::s x' or x = x' and y.::s y'. This is easily verified to be an ordering, called the lexicographic ordering; it is a total ordering whenever both 5 and T are totally ordered. References to the bibliography are by the name of the author and the date. In each section all the results are numbered consecutively, e.g. in Section 4.7 we have Theorem 4.7.1, Lemma 4.7.2, Proposition We shall also use iff as an abbreviation for 'if and only if' and. indicates the end (or absence) of a proof. Many exercises are provided with hints, and the harder ones are starred.

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