Springer Undergraduate Mathematics Series

Advisory BoardM.A.J. Chaplain, University of St. AndrewsA. MacIntyre, Queen Mary University of LondonS. Scott, King’s College LondonN. Snashall, University of LeicesterE. Süli, University of OxfordM.R. Tehranchi, University of CambridgeJ.F. Toland, University of Bath

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Jeremy Gray

A History of AbstractAlgebraFrom Algebraic Equations to Modern Algebra

123

Jeremy GraySchool of Mathematics and StatisticsThe Open UniversityMilton Keynes, UK

Mathematics InstituteUniversity of WarwickCoventry, UK

ISSN 1615-2085 ISSN 2197-4144 (electronic)Springer Undergraduate Mathematics SeriesISBN 978-3-319-94772-3 ISBN 978-3-319-94773-0 (eBook)https://doi.org/10.1007/978-3-319-94773-0

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Introduction

The conclusion, if I am not mistaken, is that above all themodern development of pure mathematics takes place under thebanner 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, itlooks at some topics in algebra and number theory and follows them from theirmodest presence in mathematics in the seventeenth and eighteenth centuries intothe nineteenth century and sees how they were gradually transformed into whatwe call modern algebra. Accordingly, it looks at some of the great success storiesin mathematics: Galois theory—the theory of when polynomial equations havealgebraic solutions—and algebraic number theory. So it confronts a question manystudents ask themselves: how is it that university-level algebra is so very differentfrom school-level algebra?

The term ‘modern algebra’ was decisively introduced by van der Waerden in hisbook Moderne Algebra (1931), and it is worth discussing what he meant by it, andwhat was ‘modern’ about it. That it still suffices as an accurate label for much of thework done in the field since is indicative of how powerful the movement was thatcreated the subject.

The primary meaning of the term ‘modern algebra’ is structural algebra: the studyof groups, rings, and fields. Interestingly, it does not usually include linear algebraor functional analysis, despite the strong links between these branches, and appliedmathematicians 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 coreconsists of the explicit solution of equations and modest excursions into geometry.The reasons for the shift in meaning, and the ways in which structural algebragrew out of old-style classical algebra, are among the major concerns of this book.Elucidating these reasons, and tracing the implications of the transformation ofalgebra, will take us from the later decades of the eighteenth century to the 1920sand 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 thatevery polynomial with real (or complex) coefficients, has as many roots as itsdegree. Because most experience with polynomial equations was tied to attemptsto solve them this problem overlapped with attempts to find explicit formulae fortheir solution, and once the polynomial had degree 5 or more no such formula wasknown. The elusive formula was required to involve nothing more than addition,subtraction, multiplication, and division applied to the coefficients of the equationand the extraction of nth roots, so it was known as solution by radicals, and Lagrangein 1770 was the first to give reasons why the general quintic equation might not besolvable by radicals.

Another source of problems in classical algebra was number theory. Fermat hadtried and largely failed to interest his contemporaries in the subject, but his writingson the subject caught the attention of Euler in the eighteenth century. Euler wroteextensively on them, and what he conjectured but could not prove was often, but notalways, soon proved by Lagrange. This left a range of partially answered questionsfor their successors to pursue. For example, Fermat had shown that odd primenumbers of the form x2 + y2 are precisely those of the form 4k + 1, and had foundsimilar theorems for primes of the form x2 + 2y2 and x2 + 3y2 but not for primesof the form x2 + 5y2—why not, what was going on? There was a good theory ofinteger solutions to the equation x2 −Ay2 = 1, where A is a square-free integer, andLagrange 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 hada conjecture about integer solutions to xn + yn = zn, n > 2 and had indeed shownthat there were no solutions when n = 4, and Euler had a suggestive but flawedproof 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 createdmodern algebraic number theory, in the sense that it inspired an unbroken stream ofleading German mathematicians to take up and develop the subject.1 The work ofGauss 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 wasnot generally solvable by radicals. This raised a deeper question, one that Gausshad already begun to consider: given that some polynomial equations of degree 5 ormore 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, andthe implications of his ideas, many of them drawn out by Jordan in 1870, also uncoilthrough the nineteenth century and figure largely in the story.

Algebraic number theory led mathematicians to the concept of commutativerings, of which, after all, the integers are the canonical example, and Galois theoryled to the concepts of groups and fields. Other developments in nineteenth centurygeometry—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 along-running argument between Kronecker and Dedekind about the proper natureof algebraic number theory. There was less disagreement about the importanceof Galois’s ideas once they were properly published, 14 years after his untimelydeath, but it took a generation to find the right way to handle them and another forthe modern consensus to emerge. By the end of the century there was a markeddisagreement in the mathematical community about the relative importance of goodquestions and abstract theory. This is not just a chicken-and-egg problem. Once it isagreed 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 andsolving them is of value in that context, but what is the point of a theory of groupswhen done for its own sake? Questions about integers may be interesting, but whatabout an abstract theory of rings? These questions acquired solid answers, ones itis the historians’ job to spell out, but they are legitimate, as are their descendantstoday: higher category theory, anyone?

The end of the nineteenth century and the start of the 20th see the shift fromclassical algebra to modern algebra, in the important sense that the structuralconcepts move from the research frontier to the core and become not only the way inwhich classical problems can be reformulated but a source of legitimate problemsthemselves. This was a lengthy process, and the publication of Moderne Algebramarks an important stage in what is still an ongoing process, one that is worththinking about. Leo Corry (1996) has usefully distinguished between what he callsthe ‘body’ and the ‘images’ of a piece of knowledge: the body of knowledge, hetells us (1996, 3) “includes theories, ‘facts’, open problems”, the images “serveas guiding principles, or selectors”, they “determine attitudes” about what is anurgent problem demanding attention, what is relevant, what is a legitimate method,what should be taught, and who has the authority to decide. It might indeed beworth separating out the social and institutional factors entirely that could be calledthe ‘forces of knowledge’: the mathematicians themselves and their institutions(universities, professional bodies, journals, and the like).

I have tried to introduce students to some of the best writing in the history ofmathematics of the last 20 or 30 years, not just for the information these booksand papers provide but as introductions to how history can be written, and to helpstudents engage with other opinions and so form their own. The most relevantexamples are the The Shaping of Arithmetic after C.F. Gauss’s DisquisitionesArithmeticae by Goldstein et al. (2007) and Corry’s Modern algebra and the riseof mathematical structures (1996), both of which in their different ways showhow much can be done when one gets away from the monotonous plod of greatachievements. Kiernan’s influential paper ‘The development of Galois theory fromLagrange to Artin’ (Kiernan 1971) is still well worth consulting. Such accounts

viii Introduction

are analytical, not merely descriptive. As Gauss famously said of his own work2

“When the building is finished the scaffolding can no longer be seen”; whatever themerits and demerits of presenting mathematics that way little is gained by treatingthe history of mathematics as an attempt to rebuild each branch of mathematics asit is today, brick by brick, with names and dates attached.

In short, this is a book on the history of algebra, and as such it asks the studentto think about what it is to study history. It emphasises some points that a straightmathematics course might marginalise, and it omits others that a mathematicianwould emphasise, as and when I judged that a history course required it. It doesassume that the students can handle difficult mathematics—and happily that provedgenerally to be the case in the years I taught it at the University of Warwick—but itrequires that they marshal arguments, based on facts, and in support of opinions, ashistorians must. I have tried to make it coherent, albeit selective.

It is not an attempt to write a ‘complete’ history of algebra in the nineteenthcentury. It is the result of a course of 30 lectures, and it can be taught as such. Notall the chapters correspond exactly to a lecture; some overshoot, and there is moreinformation in the Appendices. To reduce the study of algebra in the nineteenthcentury and the early years of the 20th to barely 300 pages meant taking some crudedecisions, and I could see no tidy way to do it. Algebra in the period, it seems, isa more heterogeneous body of knowledge than geometry was, and even real andcomplex analysis (once differential equation s of all kinds are reserved for anotheroccasion). I removed most of invariant theory and the Kleinian view of geometryfrom discussion. The history of work on determinants, matrices, linear algebragenerally, quaternions and other algebras had also to be omitted. But that only leftme facing tougher decisions. I decided not to deal with Galois’s second memoir,Sophie Germain and her work, higher reciprocity laws, Fermat’s Last Theorem,power series methods, and the distribution of prime numbers. Several of these topicsare well covered in the existing literature and would make good projects for studentswishing to take the subject further. Although it has become increasingly clear inrecent years that the work of Kummer, Kronecker, Hermite and others on the richand overlapping fields of elliptic functions, modular functions and n-ary quadraticforms was a major legacy of Gauss, none of that could be described here. In fact itmakes a suitable subject for research.3 I could not, however, resist writing about theimportance of Klein’s famous but seldom-read Lectures on the Icosahedron (1884).

No undergraduate history course assumes that the students will be competent,let alone masterful, in everything that matters to the course. Students study thetechniques of warfare without being able to ride a horse or fire a gun; diplomacywithout being taught the skills of negotiation; nationalism without engaging inpolitics; labour movements without working in a field or a factory; childhood and

2See Sartorius (1856, 82).3On Hermite, see Goldstein (2007) and Goldstein (2011); on Kronecker see Goldstein andSchappacher (2007, Section 4) and on the Kronecker–Klein dispute see (Petri and Schappacher2002).

Introduction ix

the family without becoming pregnant or rearing children. But there is a feeling thatthe history of a branch of mathematics should not be taught without the studentsacquiring something like the mastery of that branch that a straight course in it wouldhope to achieve. This book is an attempt not to break that connection—studentsshould not say nonsense about quadratic reciprocity or which equations are solvableby radicals—but to open it up to other approaches.

The ethics here are those of an applied mathematics course, in which a balanceis to be struck between the mathematics and the application. Doubtless this balanceis struck in many different ways in different courses. Here I have tried to bring outwhat is important historically in the development of this or that part of algebra, andto do so it is necessary to take some things for granted, say, that this person didvalidly deduce this result although the proof will not be looked at. The historicalpurpose in examining a strictly mathematical detail has to be decided case by case.

To give one example, perhaps the gravest in the book, the treatment of quadraticreciprocity given here contains one complete proof of the general case (Gauss’sthird proof from 1808 in which he introduced Gauss’s Lemma) and indicates howhe gave an earlier proof, in the Disquisitiones Arithmeticae, that rested on histechnique of composition of forms (although I discuss the fourth and sixth proofsin Appendix C). To explain composition of forms I indicate what is involved, andgive Dirichlet’s simplified account, but I merely hint at the difficulties involvedin securing the deepest results without which Gauss’s theory of quadratic form sloses much of its force. This gives me room to explain the history of this famouslydifficult subject, but deprives the students of some key proofs. My judgement wasthat this was enough for a history of modern algebra in which the contemporary andsubsequent appreciation of these ideas is at least as important as the accompanyingtechnicalities. I then go on to describe Dedekind’s translation of the theory into thetheory of modules and ideals that he created.

I also believe that sufficient understanding of some topics for the purposes of ahistory course is acquired by working with examples, and when these are not madeexplicit there is always an implication that examples help.

In 1817 Gauss wrote (see Werke 2, pp. 159–160) that

It is characteristic of higher arithmetic that many of its most beautiful theorems can bediscovered by induction with the greatest of ease but have proofs that lie anywhere but nearat hand and are often found only after many fruitless investigations with the aid of deepanalysis and lucky combinations

Certainly numerical exploration and verification are a good way to understandmany of the ideas in this book. That the proofs lie deep below the surface is one ofthe principal impulses for the slow elaboration of modern algebra.

x Introduction

Sources and Their Uses

Sources are at the heart of any historical account. In the last few years a huge amountof original literature in the form of journals and books has become available over theweb, and this resource will continue to grow, but inevitably much of it is in French orGerman. I have tried to note when translations are available (as they are for Gauss’sDisquisitiones Arithmeticae and all of Galois’s work) and I have added some of myown to the small collection that already exists. With a little effort more can be foundthat is in English, although that remains a problem. I have tried to indicate what wasavailable in 2017.

Historians divide sources roughly into two kinds: primary and secondary.Primary sources are the original words upon which the historian relies—here,the papers, the unpublished notes, and the letters by mathematicians themselves.Secondary sources are subsequent commentaries and other indirect records.

It will be no surprise that secondary sources for the history of mathematics onthe web are many and various. The best source for biographical information is theComplete Dictionary of Scientific Biography. The biographies on the St Andrewswebsite began as digests of this work, but are now often more up-to-date andwith interesting references to further reading. The third source of information isWikipedia. This is generally accurate on mathematical topics, but a bit more hit-or-miss on historical matters. After that it’s a lottery, and, of course, sources areoften filled with falsehoods lazily repeated, easy to find elsewhere and hard torefute. The only option is always to give your source for any significant piece ofinformation. For a website supply a working URL, for a book or article the author,title, and page number. Such references are to historians what purported proofs are tomathematicians: ways of making claims plausible or, when unsuccessful, exposingthe error.

Solid reference works in the history of mathematics are generally quite good indescribing the growth of the body of knowledge in algebra. The facts of the matterare often well known, although folk memory acts a filter: everyone knows that Gausswas a number theorist, many fewer can describe his theory of quadratic forms;everyone knows that Galois showed that the quintic is not solvable by radicals,most people guess incorrectly how he did it. The problem is that mathematicianstoday naturally suppose that the best mathematicians of the day respected thesame imperatives that they do—have the same images of algebra, one could say—and in part this is because of the successes of the structural algebra movement.Accordingly, what a history of a body of knowledge does not do well is analyseeither how the changes came about or what their implications were. The result is acuriously flat account, in which one really good idea follows another in a seeminglyinevitable way, although the development of algebra has its share of controversies,of paths not taken, and dramatic shifts of interest.

Introduction xi

Advice to Students

Throughout this book I want you, the student, to learn a number of things, each ofwhich has two aspects. I think of these as landmarks. One aspect is mathematical: Iwant you to understand and remember certain results. The other aspect is historical:I want you to understand the importance of those results for the development ofmathematics.

In each case you will have to do some preliminary work to get where I wantyou to be. You will have to absorb some mathematical terms and definitionsin order to understand the result and have a good idea about its proof. Youwill also have to absorb some sense of the intellectual enterprise of the mathe-maticians and their social context to appreciate the historical dimension of theirresults.

I have listed some key results below, almost in the order in which they occur inthe book. A good way to test your understanding as your study progresses would beto ask yourself if you have understood these results both as pieces of mathematicsand as turning points in the history of mathematics. Understanding is a relative term;aim for knowing the definitions and the best examples, the key steps in the proofs ofthe main theorems and, equally important, a sense of the context in which the resultwas first proposed and what was significant about it.

All these concepts will be taught as if they were new, although some familiaritywith the more elementary ones will be helpful; these include the basic definitionsand theorems in group theory and ring theory. The topic of continued fractions andPell’s equation is mentioned but marginalised here; it would make an interestingproject to go alongside what I have presented. The same is true of the Kleinianview of geometry, which is an important source of group-theoretic ideas andcontributed significantly to the recognition of their importance. On the other hand, Iassume no previous acquaintance with algebraic number theory (algebraic numbers,algebraic integers, algebraic number fields, quadratic and cyclotomic fields, ideals,the factorisation of algebraic integers into irreducibles, principal ideal domains, andthe prime factorisation of ideals) nor with a modern course in Galois theory, whichlooks at the subject in more detail and from a more modern abstract perspective thanwe do here.

The treatment of the mathematics in this book is surely not the modern one, ratherit is one closer to the original, but I believe that the formulations of a Dedekind or aJordan have a lucidity and a naturalness that commends them, and that in presentingthem it is possible to teach the mathematics through its history and in so doing toteach its history in a full and interesting way. In particular, it becomes possible tounderstand that substantial ideas in modern mathematics were introduced for goodand valuable reasons.

Any living body of knowledge is different from what is revealed by an autopsy.Research mathematicians see problems, methods, and results, organised perhaps byan image of mathematics, governed also by a sense of their own abilities. They haveproblems they want to solve, which can be of various kinds, from just getting an

xii Introduction

answer to getting a known answer by a better method. They can be theory builders,example finders, or problem solvers—all three at different times. What a study of thehistory of mathematics can provide, albeit for the past, is the dynamic appreciationof mathematics where research questions, their possible solutions, their difficultyand importance are always visible.

This book, at least, has two sorts of aims. One is to describe some of themajor steps on the journey from classical algebra to modern algebra. The otheris to describe how some mathematicians have come to formulate, and even solve,important new problems in mathematics. These two aims are more independent thanit might seem at first glance. Firstly, the advances made by the mathematicians werenot aimed at creating modern algebra until the journey was more or less complete.So we might well want to think of that journey as proceeding at random, in anaimless way, and certainly not as the result of a design held for over a century orlonger. Secondly, the way research tasks get formulated and even solved is largelyindependent of the over-arching story into which they can be made to fit. A goodmathematician has a good grasp of (much of) what is known in a subject, and away of organising this knowledge so that it generates questions. Then, with luck,he or she answers those questions, or, in failing, answers others that might, withhindsight, seem better. Again, there is a random element here. The new insight is,or is not, picked up by others and incorporated into their view of the subject, and sothe whole creaky edifice advances.

We can think of a researcher (in any subject, I would think, not just mathematics)having a story they tell themselves about their part of the subject. This story hasdefinitions, concepts, and theorems, and it culminates in things the mathematicianwould like to know, which may or may not be what other mathematicians wouldlike to know. We can call this the local or personal story. The historian, orthe historically-minded mathematician, fits some aspects of these local storiestogether to get a global story—such as the origins of modern algebra. Mostlikely the historian retains more of the personal stories and the mathematicianmore of the results, probably re-writing them in modern terms as he or sheproceeds.

So understanding this book is a matter of appreciating the local stories, andallowing them to fit together in some way. This determines what your first questionshave to be: What did this person know about the topic? What did he or she wantto know? How do they think they could do it? If you like, think of simply gettingto know several mathematicians on their own terms, without fitting them into somehistorical account you may have heard before.

To take these questions in order, the first one does involve you understandingthe mathematics the mathematician understood. Perhaps the subject is new to you,perhaps you know it already in a more modern form. I advise resisting the temptationto tackle all the mathematics, but to give way sometimes. After all, you exercise thesame judgements when finding out about many interesting things, and you haveother things to do here, too. One thread to hold on to is provided by examples,rather than theorems and proofs, and another is to able to describe the significanceof a result (rather than merely repeat it).

Introduction xiii

The second question is perhaps simpler. The answer will be of the form: Solvethis problem; Prove this theorem; Formulate some general principle. Again, thesignificance of these answers is as important as the answers themselves; what chancedid this or that problem have of engaging the interest of anyone else?

The third question is more about mathematical methods, but here we want tonotice a few things. These mathematicians did not always succeed: not completely,and sometimes not at all. This highlights one difference between this book andsome histories of mathematics: it is not about who first did what. It is about whatpeople were trying to do, and the extent to which they succeeded. Success cantake many forms, not all of them welcome: the counter-example, the refutation, theunexpected example are all advances, but they may not fit the picture people wantedto create.

In most of the book each chapter has some exercises of a mathematical kindto help you grapple with the concepts, and some questions of a more historicalnature to point you in the various directions that might help you contextualise themathematics and reveal how it was evolving. In the final third of the book theexercises fall away, because I believe that the conceptual shifts towards modernmathematics are in fact clearer and easier to see from a slight distance than in theclose-up view that exercises provide.

When we look at the stories about the discovery of this or that piece ofmathematics and its path to acceptance, we also want to fit them into the globalstory we have chosen. Here we must be careful. Many historians would objectto an enterprise focussed on how some aspect of the modern world came intobeing, finding the question too likely to lead to tendentious answers that lose toomuch historical diversity. It is true that it is hard enough, and interesting enough,to say accurately how and why Lagrange, Gauss, Galois or who ever did whatthey did without imagining that we can explain how some part of the modernworld came about. But I think that the arrival of modern algebra is itself clearlya historical event, and it is plausible to explain it in historical terms. We can dothis without supposing that everything was oriented to some great event in thefuture.

There is a particular issue to do with how you handle information, especiallywhen it might be wrong. To repeat a historical falsehood (say, that Jacobi wasa Professor in Berlin all his working life) is a mistake (Jacobi was in fact atKönigsberg for most of his career), but it doubly misleading if it is relayed as animportant fact without a source.

But the more important distinction is between facts and opinions. The Jacobierror is an error of fact. If you say, however, that Klein’s Erlangen programmeof 1872 was very important for the growth of group theory, that’s an opinion. Itmay be true—I think it isn’t—but even if you want to argue that it is a validopinion, it is certainly an opinion. When you find such claims in the literature, ifthey matter to your essay you have to cite them. You do that by writing somethinglike “As Smith has argued (Smith 1972, p. 33) . . . ”, or, if you disagree, “Contraryto what Smith has argued (Smith 1972, p. 33) . . . .” In each case (Smith 1972,p. 33) refers to an item in your references by Smith (with an initial or first name)

xiv Introduction

published in 1972 (give the title and tell me how I can find it) in which thisopinion is on page 33. You may also write “Smith (1972, 33) argued that . . . ”;in this case the subject is Smith the person, and the parenthetical reference isto an item in your bibliography. You do the same with quotations, of course.If the item is on the web, you give the URL, and a page reference if that ispossible.

The fact–opinion distinction isn’t easy to maintain (when does a widely sharedopinion simply become a fact?) but it matters because there is not much you cando with a well-established fact. Unless you suspect that it is wrong, and researchin the archives will refute it, you must accept it. But an opinion is an opinion.Here’s one of mine: Dirichlet, who was a friend of Jacobi’s and did teach inBerlin for many years, was hugely influential in bringing rigorous mathematics toGermany and making Berlin in particular and Prussia generally a major centre formathematics. You can see it’s an opinion: I used the words “hugely influential”and “major centre”. My evidence would have to be a major shift in the quality ofwork done in mathematics in Prussia, most of it demonstrably linked to Dirichletdirectly or through the force of his example, and a clear rise of Berlin to a positionof international eminence.

This book is full of opinions, and I expect you to have yours. Even if we agree youshould have arguments in support of our opinions, and these arguments might notbe the same. Certainly if we disagree we should both have arguments. We can thenprobe each other’s positions, shift our opinions, refine them or agree to disagree. Butthe fact–opinion distinction is central to this book, so if you are giving an opiniontaken from somewhere, say where it comes from.

Can you have too many references? In principle, yes. Will you, unless you setout deliberately to do so—no!

I have provided a number of appendices. Some amplify the historical story, andcould be used instead of some of the later chapters of this book or as introductionsto projects. Some—those involving Gauss, Dedekind, and Jordan—also containtranslations of original material. One, on permutation groups, revises some materialthat I did not want to assume or to teach historically, and it can be used as a reference.

Acknowledgements

I thank John Cremona and Samir Siksek for their encouragement. I thank the variousreferees for the comments, particularly the anonymous referee who went to thetrouble to inform me of a great many misprints; every reader owes him a debtfor performing that valuable service. And I thank Anne-Kathrin Birchley-Brun atSpringer for her expert editorial assistance and advice.

Introduction xv

Results Discussed in This Book

1. The Tartaglia–Cardano method for solving a cubic equation and the formulathat results; a method for attacking polynomial equations of degree 4;

2. the characterisation of numbers of the form x2 + y2, x2 + 2y2, x2 + 3y2, andwhat is different about numbers of the form x2 + 5y2;

3. Pell’s equation x2 −Ay2 = 1 for square-free integers A and its connection withthe continued fraction expansion of

√A;

4. Fermat’s last theorem, an indication of his proof for 4th powers and of Euler’s‘proof’ for 3rd powers;

5. Lagrange’s reduction of binary quadratic forms to canonical form, especially incases of small positive discriminants;

6. the statement of the theorem of quadratic reciprocity and its use in simple cases;7. a proof of the theorem of quadratic reciprocity, and an understanding of why

Gauss gave several proofs;8. cyclotomic integers and an unsuccessful connection to Fermat’s Last Theorem;9. the key features of the Gauss–Dirichlet theory of quadratic forms;

10. the quintic equation is not solvable by radicals, with an indication of Abel’sproof and Galois’s approach;

11. it is impossible to trisect an angle or double a cube by straight edge and circlealone;

12. Galois’s theory of solvability by radicals, as explained by Galois, Jordan, Klein,and Weber;

13. Cauchy’s theory of subgroups of Sn;14. transitivity and transitive subgroups of Sn;15. the field-theoretic approaches to Galois theory of Kronecker and Dedekind;16. algebraic integers; ideals and prime ideals in a ring of algebraic integers;17. ideals in rings of quadratic integers and the connection to classes of binary

quadratic forms;18. the Hilbert basis theorem;19. the Brill–Noether AF + BG theorem, the counter-example provided by the

rational quartic in space;20. primary ideals, primary decomposition in a commutative ring.

Contents

1 Simple Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Pell’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Fermat’s Proof of the Theorem in the Case n = 4 . . . . . . . . . . . . . . . . . . 152.3 Euler and x3 + y3 = z3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Lagrange’s Theory of Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The Beginnings of a General Theory of Quadratic Forms . . . . . . . . . 233.3 The Theorem of Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Taking Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Gauss’s Disquisitiones Arithmeticae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 The Disquisitiones Arithmeticae and Its Importance . . . . . . . . . . . . . . . 384.3 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Gauss on Congruences of the Second Degree . . . . . . . . . . . . . . . . . . . . . . 424.5 Gauss’s Theory of Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Cyclotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The Case p = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 The Case p = 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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6 Two of Gauss’s Proofs of Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . 576.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Composition and Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.3 Smith’s Commentary on Gauss’s Sixth Proof. . . . . . . . . . . . . . . . . . . . . . . 616.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Dirichlet’s Lectures on Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Gauss’s Third Proof of Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . 677.3 Dirichlet’s Theory of Quadratic Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.4 Taking Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8 Is the Quintic Unsolvable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2 Solution of Equations of Low Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.3 Lagrange (1770) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.5 Revision on the Solution of Equations by Radicals . . . . . . . . . . . . . . . . 92

9 The Unsolvability of the Quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.2 Ruffini’s Contributions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.3 Abel’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.4 Wantzel on Two Classical Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049.5 Wantzel on the Irreducible Case of the Cubic. . . . . . . . . . . . . . . . . . . . . . . 1089.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

10 Galois’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11510.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11510.2 Galois’s 1st Memoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12010.3 From Galois’s Letter to Chevalier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.5 A Cayley Table of a Normal Subgroup.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13010.6 Galois: Then, and Later . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11 After Galois . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.2 The Publication of Galois’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13311.3 Serret’s Cours d’Algèbre Supérieure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13611.4 Galois Theory in Germany: Kronecker and Dedekind . . . . . . . . . . . . . 138

12 Revision and First Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

13 Jordan’s Traité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14913.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14913.2 Early Group Theory: Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15013.3 Jordan’s Traité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

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13.4 Jordan’s Galois Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15413.5 The Cubic and Quartic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

14 The Galois Theory of Hermite, Jordan and Klein . . . . . . . . . . . . . . . . . . . . . . 16314.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16314.2 How to Solve the Quintic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16314.3 Jordan’s Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16814.4 Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17114.5 Klein in the 1870s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17214.6 Klein’s Icosahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17414.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

15 What Is ‘Galois Theory’? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17915.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17915.2 Klein’s Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18015.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

16 Algebraic Number Theory: Cyclotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.2 Kummer’s Cyclotomic Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.3 Fermat’s Last Theorem in Paris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

17 Dedekind’s First Theory of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19517.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19517.2 Divisibility and Primality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19517.3 Rings, Ideals, and Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19817.4 Dedekind’s Theory in 1871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

18 Dedekind’s Later Theory of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20318.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20318.2 The Multiplicative Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20318.3 Dedekind and ‘Modern Mathematics’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20618.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

19 Quadratic Forms and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20919.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20919.2 Dedekind’s 11th Supplement, 1871–1894 . . . . . . . . . . . . . . . . . . . . . . . . . . 20919.3 An Example of Equivalent Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

20 Kronecker’s Algebraic Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21720.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21720.2 Kronecker’s Vision of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21720.3 Kronecker’s Lectures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22320.4 Gyula (Julius) König . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

21 Revision and Second Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

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22 Algebra at the End of the Nineteenth Century . . . . . . . . . . . . . . . . . . . . . . . . . . 23522.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23522.2 Heinrich Weber and His Textbook of Algebra . . . . . . . . . . . . . . . . . . . . . . . 23522.3 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23822.4 Number Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

23 The Concept of an Abstract Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24523.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24523.2 Moore, Dickson, and Galois Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24523.3 Dedekind’s 11th Supplement, 1894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24823.4 Kürschák and Hadamard .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24923.5 Steinitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

24 Ideal Theory and Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25524.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25524.2 The Brill–Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25624.3 The Failure of the Brill–Noether Theorem to Generalise . . . . . . . . . . 25824.4 Lasker’s Theory of Primary Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25924.5 Macaulay’s Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26124.6 Prime and Primary Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

25 Invariant Theory and Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26325.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26325.2 Hilbert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26325.3 Invariants and Covariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26725.4 From Hilbert’s Paper on Invariant Theory (1890) . . . . . . . . . . . . . . . . . . 26725.5 The Hilbert Basis Theorem and the Nullstellensatz . . . . . . . . . . . . . . . . 272

26 Hilbert’s Zahlbericht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27526.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27526.2 An Overview of the Zahlbericht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27526.3 Ideal Classes and Quadratic Number Fields . . . . . . . . . . . . . . . . . . . . . . . . 27726.4 Glimpses of the Influences of the Zahlbericht . . . . . . . . . . . . . . . . . . . . . . 279

27 The Rise of Modern Algebra: Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 28127.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28127.2 The Emergence of Group Theory as an Independent Branch

of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28127.3 Dickson’s Classification of Finite Simple Groups .. . . . . . . . . . . . . . . . . 286

28 Emmy Noether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28928.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29028.2 Ideal Theory in Ring Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29228.3 Structural Thinking.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

29 FromWeber to van der Waerden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29729.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29729.2 van der Waerden on the Origins of Moderne Algebra . . . . . . . . . . . . . . 302

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30 Revision and Final Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

A Polynomial Equations in the Eighteenth Century . . . . . . . . . . . . . . . . . . . . . . 309A.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309A.2 The Fundamental Theorem of Algebra Before Gauss . . . . . . . . . . . . . . 309

B Gauss and Composition of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319B.1 Composition Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319B.2 Gaussian Composition of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323B.3 Dirichlet on Composition of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325B.4 Kummer’s Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

C Gauss’s Fourth and Sixth Proofs of Quadratic Reciprocity . . . . . . . . . . . 331C.1 Gauss’s Fourth Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331C.2 Gauss’s Sixth Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333C.3 Commentary .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

D From Jordan’s Traité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343D.1 Jordan, Preface to the Traité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343D.2 Jordan, General Theory of Irrationals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345D.3 Jordan: The Quintic Is Not Solvable by Radicals . . . . . . . . . . . . . . . . . . . 361D.4 Netto’s Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

E Klein’s Erlanger Programm, Groups and Geometry . . . . . . . . . . . . . . . . . . . 367E.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367E.2 Felix Klein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367E.3 Geometric Groups: The Icosahedral Group .. . . . . . . . . . . . . . . . . . . . . . . . 370E.4 The Icosahedral Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

F From Dedekind’s 11th Supplement (1894) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

G Subgroups of S4 and S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389G.1 The Subgroups of S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389G.2 The Subgroups of S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

H Curves and Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393H.1 Intersections and Multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

I Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397I.1 Netto’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397I.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

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J Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401J.1 Other Accounts of the History of Galois Theory . . . . . . . . . . . . . . . . . . . 401J.2 Other Books on the History of Algebraic Number Theory .. . . . . . . . 402

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

List of Figures

Fig. 4.1 Carl Friedrich Gauss (1777–1855). Image courtesy ofDr. Axel Wittmann, copyright owner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Fig. 9.1 Niels Henrik Abel (1802–1829). Photo courtesy of theArchives of the Mathematisches ForschungsinstitutOberwolfach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Fig. 9.2 Descartes’s solution to a quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . 105Fig. 9.3 Augustin Louis Cauchy (1789–1857) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Fig. 10.1 Évariste Galois (1811–1832) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Fig. 10.2 Scholarly Paris in 1830 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Fig. 11.1 Leopold Kronecker (1823–1891). Photo courtesy ofthe Archives of the Mathematisches ForschungsinstitutOberwolfach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Fig. 13.1 Camille Jordan (1838–1922) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Fig. 14.1 Charles Hermite (1822–1901). Photo courtesy of theArchives of the Mathematisches ForschungsinstitutOberwolfach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Fig. 14.2 Christian Felix Klein (1849–1925) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Fig. 18.1 Richard Dedekind (1831–1916). Photo courtesy of theArchives of the Mathematisches ForschungsinstitutOberwolfach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Fig. 20.1 The three real inflection points of the curvex(1 − x2) = y(y2 − x2) lie on the x-axis . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Fig. 22.1 Heinrich Martin Weber (1842–1913). Photo courtesy ofthe Archives of the Mathematisches ForschungsinstitutOberwolfach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Fig. 25.1 David Hilbert (1862–1943). Photo courtesy of the Archivesof the Mathematisches Forschungsinstitut Oberwolfach.. . . . . . . . . . . 263

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Fig. 28.1 Amalie Emmy Noether (1882–1935). Photographic portraitof Emmy Noether, Photo Archives, Special CollectionsDepartment, Bryn Mawr College Library . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Fig. E.1 A square in a dodecahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373Fig. E.2 Relabelling the faces of a dodecahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Fig. H.1 The folium of Descartes, x3 + y3 = 3xy . . . . . . . . . . . . . . . . . . . . . . . . . . . 394