Springer Undergraduate Mathematics Series Jeremy Gray A History of Abstract Algebra From Algebraic Equations to Modern Algebra Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain, University of St. Andrews A. MacIntyre, Queen Mary University of London S. Scott, King’s College London N. Snashall, University of Leicester E. Süli, University of Oxford M.R. Tehranchi, University of Cambridge J.F. Toland, University of Bath More information about this series at http://www.springer.com/series/3423 Jeremy Gray A History of Abstract Algebra From Algebraic Equations to Modern Algebra 123 Jeremy Gray School of Mathematics and Statistics The Open University Milton Keynes, UK Mathematics Institute University of Warwick Coventry, UK ISSN 1615-2085 ISSN 2197-4144 (electronic) Springer Undergraduate Mathematics Series ISBN 978-3-319-94772-3 ISBN 978-3-319-94773-0 (eBook) https://doi.org/10.1007/978-3-319-94773-0 Library of Congress Control Number: 2018948208 Mathematics Subject Classification (2010): 01A55, 01A60, 01A50, 11-03, 12-03, 13-03 © Springer Nature Switzerland AG 2018 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, express 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 The conclusion, if I am not mistaken, is that above all the modern development of pure mathematics takes place under the banner of number. David Hilbert,The Theory of Algebraic Number Fields, p. ix. Introduction to the History of Modern Algebra This book covers topics in the history of modern algebra. More precisely, it looks at some topics in algebra and number theory and follows them from their modest presence in mathematics in the seventeenth and eighteenth centuries into the nineteenth century and sees how they were gradually transformed into what we call modern algebra. Accordingly, it looks at some of the great success stories in mathematics: Galois theory—the theory of when polynomial equations have algebraic solutions—and algebraic number theory. So it confronts a question many students ask themselves: how is it that university-level algebra is so very different from school-level algebra? The term ‘modern algebra’ was decisively introduced by van der Waerden in his book Moderne Algebra (1931), and it is worth discussing what he meant by it, and what was ‘modern’ about it. That it still suffices as an accurate label for much of the work done in the field since is indicative of how powerful the movement was that created the subject. The primary meaning of the term ‘modern algebra’ is structural algebra: the study of groups, rings, and fields. Interestingly, it does not usually include linear algebra or functional analysis, despite the strong links between these branches, and applied mathematicians may well encounter only a first course in groups and nothing else. Modern algebra is in many ways different from school algebra, a subject whose core consists of the explicit solution of equations and modest excursions into geometry. The reasons for the shift in meaning, and the ways in which structural algebra grew out of old-style classical algebra, are among the major concerns of this book. Elucidating these reasons, and tracing the implications of the transformation of algebra, will take us from the later decades of the eighteenth century to the 1920s and the milieu in which Moderne Algebra was written. v vi Introduction Classical algebra confronted many problems in the late eighteenth century. Among them was the so-called fundamental theorem of algebra, the claim that every polynomial with real (or complex) coefficients, has as many roots as its degree. Because most experience with polynomial equations was tied to attempts to solve them this problem overlapped with attempts to find explicit formulae for their solution, and once the polynomial had degree 5 or more no such formula was known. The elusive formula was required to involve nothing more than addition, subtraction, multiplication, and division applied to the coefficients of the equation and the extraction of nth roots, so it was known as solution by radicals, and Lagrange in 1770 was the first to give reasons why the general quintic equation might not be solvable by radicals. Another source of problems in classical algebra was number theory. Fermat had tried and largely failed to interest his contemporaries in the subject, but his writings on the subject caught the attention of Euler in the eighteenth century. Euler wrote extensively on them, and what he conjectured but could not prove was often, but not always, soon proved by Lagrange. This left a range of partially answered questions for their successors to pursue. For example, Fermat had shown that odd prime numbers of the form x2 + y2 are precisely those of the form 4k + 1, and had found similar theorems for primes of the form x2 + 2y2 and x2 + 3y2 but not for primes of the form x2 + 5y2—why not, what was going on? There was a good theory of integer solutions to the equation x2 −Ay 2 = 1, where A is a square-free integer, and Lagrange had begun a theory of the general binary quadratic form, ax2 +bxy+cy2, but much remained to be done. And, famously today if less so in 1800, Fermat had a conjecture about integer solutions to xn + yn = zn,n > 2 and had indeed shown that there were no solutions when n = 4, and Euler had a suggestive but flawed proof of the case x3 + y3 = z3 that led into the theory of quadratic forms. The nineteenth century began, in algebra, with Gauss’s Disquisitiones Arithmeti- cae (1801), the book that made Gauss’s name and may be said to have created modern algebraic number theory, in the sense that it inspired an unbroken stream of leading German mathematicians to take up and develop the subject.1 The work of Gauss and later Dedekind is central to the story of the creation of modern algebra. In the 1820s Abel had wrapped up the question of the quintic, and shown it was not generally solvable by radicals. This raised a deeper question, one that Gauss had already begun to consider: given that some polynomial equations of degree 5 or more are solvable by radicals, which ones are and how can we tell? It was the great, if obscure, achievement of Galois to show how this question can be answered, and the implications of his ideas, many of them drawn out by Jordan in 1870, also uncoil through the nineteenth century and figure largely in the story. Algebraic number theory led mathematicians to the concept of commutative rings, of which, after all, the integers are the canonical example, and Galois theory led to the concepts of groups and fields. Other developments in nineteenth century geometry—the rediscovery of projective geometry and the shocking discovery of 1Gauss also rediscovered the first known asteroid in 1801. Introduction vii non-Euclidean geometry—also contributed to the success of the group concept, when Klein used it in the 1870s to unify the disparate branches of geometry. The rise of structural mathematics was not without controversy. There was a long-running argument between Kronecker and Dedekind about the proper nature of algebraic number theory. There was less disagreement about the importance of Galois’s ideas once they were properly published, 14 years after his untimely death, but it took a generation to find the right way to handle them and another for the modern consensus to emerge. By the end of the century there was a marked disagreement in the mathematical community about the relative importance of good questions and abstract theory. This is not just a chicken-and-egg problem. Once it is agreed that theory has a major place, it follows that people can work on theory alone, and the subject has to grow to allow that. By and large, equations have contexts and solving them is of value in that context, but what is the point of a theory of groups when done for its own sake? Questions about integers may be interesting, but what about an abstract theory of rings? These questions acquired solid answers, ones it is the historians’ job to spell out, but they are legitimate, as are their descendants today: higher category theory, anyone? The end of the nineteenth century and the start of the 20th see the shift from classical algebra to modern algebra, in the important sense that the structural concepts move from the research frontier to the core and become not only the way in which classical problems can be reformulated but a source of legitimate problems themselves.