Galois Theory itself is only one sub-division of Group Theory, while Group Theory is only one aspect (albeit the most basic) of the huge field of Modern Algebra. One is dealing with a 3-fold evolution, with each subject nested inside the other, sometimes helping, sometimes even hindering the others along the path to fulfillment. A brief chronology will be followed by a more detailed reconstruction of the main points:
Evariste Galois brought together and clarified the underlying ideas of Lagrange, Abel, and Fubini , who, each in his own way, showed that not all quintic polynomials over the rationals have roots that can be expressed in terms of radicals. He did his work in the period 1829-1832. His scattered papers (the expression used to describe them was un gachis ) were brought together by his friend Auguste Chevalier and deposited with the French Academy of Sciences.
Nothing was done with them for 11 years, until the famous mathematician Joseph Liouville was persuaded to look them over. In 1846 they were published in Liouville's journal of pure and applied mathematics. Thus, 14 years had passed before the world had a chance to study them.
The first person to extend Galois' results was the Italian Enrico Betti (known for the Betti numbers in Topology) in some papers written in 1851 and 1852. His knowledge of group theory was flawed; in this respect he resembled most mathematicians. At the same time he contributed some important new concepts.
After 1852 there appears to have been no research based on Galois's ideas until the 1870's. The most notable British mathematicians, Sylvester and Cayley, read his papers and found them incomprehensible. England would contribute nothing to the Galois revolution in mathematics until the 1890's. However, at the same time its mathematicians were advancing modern algebra and group theory along very different lines, with which the public is very familiar today: Boolean Algebra, the mathematical foundation of the computer age; and the study of symmetry principles in geometry.
Although Galois' ideas were not being used, they were studied by German mathematicians, and explained in textbooks. In 1854, 6 years after Liouville's publication of Galois Nachlass , Leopold Kronecker's textbook on algebra, which treats the work of Galois, was translated into French. This effectively brought Galois Theory back to its country of origin.
In 1856 Galois' "Proposition 2" was treated in the textbook of Richard Dedekind, the founder, in some sense, of Abstract Algebra. It is in this proposition that Galois defines the concept of a group; the concept is not axiomatized, yet is essentially what we mean by the concept of a group today.
Unlike the situation in Galois' case, Dedekind's impact was immediate and far-reaching. Emmy Nöther ( the greatest woman research mathematician in the standard model of the history of mathematics ) is reputed to have said that every problem in abstract algebra can be resolved by searching the works of Dedekind. Richard Dedekind also did work on the theory of fields, from 18975 to 1880. Groups and fields are intimately related in Galois Theory.
The first French algebra textbook to speak, somewhat parenthetically, about Galois' discoveries was that of J.A. Serret, published in 1849. In the third edition of 1866, he uses Galois' methods in discussing practical ways of finding roots of equations, but does not introduce the group concept. This book would eventually prove to be a damper on research in algebra in France. Serret's textbook was very popular; in fact it became the standard textbook on the subject in the French university system, going through more than a dozen editions from 1866 to 1928! However it appears that no new material was added to it for 60 years, seriously inhibiting progress in this field in France. The hostility of French science to all things German, dating to 1870, must have discouraged the majority of mathematicians from learning German. It should be pointed out however that the American tradition in mathematics, in both research and teaching, was established by graduate students who traveled to Germany in the 19th century to study with Felix Klein.
Given this environment, it is all the more surprising that two French mathematicians were, all the same, able to make notable advances in both group theory and Galois theory. These were Camille Jordan and Charles Hermite. In 1870 Jordan compiled a textbook that covered everything that was known in group theory to that time. It was, in fact, the first textbook to treat group theory as an autonomous subject of pure mathematics, unrelated to any of its practical applications.
The first textbook to relate the connection between groups and fields that is central to modern Galois theory, was that of Heinrich Weber in 1893. However, from that date to the definitive reformulation of Galois theory in the language of abstract algebra, published by Emil Artin in 1933, no further research was done at all in Galois theory.
This chronology, established through a series of lectures on Tuesday and Wednesday October 25 and 26th , was summed up in the talk of Colin McClarty. The creation of modern Galois Theory, he explained, was done in Göttingen, Germany: Emmy Nöther, Emil Artin, Zassenhaus, Hasse, building on the ideas of Dedekind, Kronecker and Weber.
On the morning of the final day of the conference, October 28, J.P. Serre recapitulated this chronology, (with changes of emphasis):
1800: Gauss fundamental theorem of algebra, quadratic reciprocity, cyclotomic equations
1830 Galois, Abel, Jacobi, Fubini
1837 Dirichlet, Analytic Methods, L-functions
1844 Eisenstein Cubic Reciprocity
1859 Riemann . Riemann specifically credited Galois for inspiring his own investigations into prime numbers
1896 Prime Number Theorem
1917 Hecke, new L-functions
1923, 1933 Emil Artin
One can identify the factors in 19th century history that contributed to this phenomenon. In previous centuries it was taken for granted that an artist or scientist had to be affiliated with a court, a university or the church. With the exponential advances of the Industrial Revolution ( and its accompanying political revolutions) , it was recognized by many intelligent young people that it was possible to succeed, or accomplish important things, in the arts and sciences without royal patronage or a fixed position in the religious or academic establishments. But such work had to be undertaken on faith: there was little hope of finding publishers, or being treated with the respect due to one's abilities, or even of surviving with a decent income.
Today it is understood that major achievements in the arts are possible for persons from every level of society. This idea has yet to make headway in the sciences, which are still dominated by the shadow of the academy. At the same time, the means for making a decent income (at least in the Western world) have greatly diversified. It was possible for TS Eliot to be a bank clerk, or Ravel to be internationally famous even without the imprimatur of the reactionary Paris Conservatory, that never did give him a degree after 12 years of hanging on there . James Joyce could survive for years as a language teacher for Berlitz, and Wallace Stevens work in an insurance company. Charles Ives could even found an insurance company! Albert Einstein himself did and published his epoch-making research while employed as a patent clerk in Bern.And even Ezra Pound could find work as a radio broadcaster (sick joke ).
These things were not impossible in previous centuries, but they were very very rare (Spinoza with his lenses). For the most part, original thinkers without stable patronage from the upper elites did not, or could not, have the option of dedicating themselves to their work, certainly not of promoting it. Indeed the promotion, in almost all cases had to be done by others: as a classic example, Beethoven's brother gave up his own career to spend full time promoting his compositions.
This being the case it was inevitable that recognition would be long in coming, the transmission from decade to decade sporadic, with many reverses, desperate rescues, and constant danger that their work would disappear into oblivion.
van der Waerden's classic textbook of the 1930's was mentioned as a kind of official 'benchmark' for the creation of 'modern algebra' as an autonomous discipline. By setting up a 'schematic picture' of algebraic structures, this textbook brought together the basics of all of its branches .
In terms of my personal interests the most valuable of these lectures was that which opened the Wednesday morning session:
"Renaud Chorlay: Groupes de symmétrie en géomètrie. Renouvellement des perspectives dans les années 1920"
Chorlay described the strong alienated that separated the proponents of Riemannian geometry, which is based on differential metric forms, and the Erlangen Programme enunciated by Felix Klein in 1872 , which had as its goal the description and classification of all geometries in terms of group theoretic symmetries.
The basic difficulty which obstructed the effective symbiosis of these two approaches was the fact that the phenomenon of "parallel transport" in differential geometry is not "open": if a vector is moved by parallel transport around a closed loop in a Riemannian space, the vector at the end of the voyage may be pointing in a different direction from the initial vector.
The essence of the group structure , however, is that it is operatonally closed: operations by elements on other elements produce a third element, all of which remain within the group. This, combined with the existence of inverses and an identity, seems to make the group model unsuitable for Riemannian geometry. Yet the Erlangen Program maintained that it could characterize all geometries.
The problem was resolved in the 20's through the introduction of the concept of the "infinitesimal group" from Sophus Lie's theory of differential groups and algebras. Lie cited Galois as a basic inspiration for his research, though the resultant theory is very different. In 1924, Elie Cartan revised Klein's Erlangen Programme to include Lie groups. In 1921 Hermann Weyl built an interpretation of Relativity based entirely on Group Theory in his treatise Raum-Zeit-Materei . This famous book presents an axiomatic treatment of Riemannian geometry using group actions, metric fields and affine connections.
In 1928, Oswald Veblen published his textbook: "Differential Invariants and Geometry" which covers all the advances in differential geometry made up to then.
Finally in 1946, Claude Chevalley published (in English) the immensely popular textbook "Theory of Lie Groups", which has gone through 15 editions. From the blurb on the Amazon website: "This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms."
In the same period that the Galois conference was happening, there was a (magnificent) exhibition at the Musée du Grand-Palais of the paintings of the collections of Leo and Gertrude Stein from the 20's and 30's. I dashed off one afternoon to attend a conference at IRCAM on music, category theory and epistemology. The Village Voice bookstore hosted the Faulkner scholar, Phillip Weinstein , for a book signing of the translation of his book "Becoming Faulkner". There was a concert of contemporary music from Ireland at the Irish Cultural Center (less than a block away from the IHP) in which personal acquaintances were performing . And dinners with friends, old and new. And playing folk tunes on a recorder outside the doors of Shakespeare and Company, by which I expedient I picked up about 30 needed dollars, my only income from 11 weeks in Europe. And..
I will therefore end this report with a bouquet of Assorted Topics, mostly human interest material relating to mathematical events in Paris at that time.
Animath is an organization with its offices on the 4th floor of the IHP building. It invents and coordinates events designed to arouse and encourage interest in mathematics in secondary education. On this occasion the audience was there to listen to a talk by the algebraic geometer, Pierre Cartier, on the life and work of Evariste Galois.
When they'd all assembled and quieted down, the introductories were was given by the director of Animath, Martin Andler. Andler speaks English so fluently that it is reasonable to conjecture that he's spent several years in the US, probably at universities. Lively, brisk, wearing the standard French formal blue educator suit, very much the dedicated teacher . In addressing the student assembly Andler evoked the "mythical environment"of the Poincaré Institute. Quote:
"Hundreds of mathematicians have passed through these doors". The same of course could be said of Customs at the Charles de Gaulle airport; but we all know what he means. In the period from 1920 to 1960 it was the principal math research institute in France. After 1960 this role was picked up by the Institut des Hautes Etudes Scientifiques . In recent years it has again become a major conference center for mathematics researchers. Once Andler stepped off the podium the lively buzz of chatter of a high school assembly started up immediately; it quieted down again at the arrival of the director of the IHP, Cedric Villani.
Cedric Villani strikes quite a different figure from Marc Andler. This was hardly surprising: Villani's sartorial originality is not likely to find imitators. As always, he wore a dark blue suit/uniform right out of the 19th century, with a hair-style to match. I've never seen him dressed otherwise. Draped around his neck was a grand maroon scarf.(Sometimes he wears a flamboyant maroon "foulard") His face was intense and withdrawn, with dark rings around cavernous eyes.
I should talk: in the exciting years when I lived in Berkeley, California (1983-87), I regularly performed music and recited poetry in Sproul Plaza on the UC Berkeley campus.This was complemented by appropriate costuming, artistic, lyrical and often loud. When heading over to Evans Hall to attend the Thursday afternoon mathematics colloquium I never bothered to change back into more respectable academic garb before entering the auditorium. This could, and did, cause outrage and panic from some of the more insecure savants. Don't you believe that stuff about the anarchist non-conformity of UC Berkeley. Nowadays one only finds it among the homeless street people living and sleeping down Telegraph Avenue.
Villani went on to say that scientists are not celebrities, like movie stars and politicians, yet their work is very important. He himself became a celebrity, briefly, in 2008, when he won the Fields Medal. People would approach him in the Metro and ask for his autograph!
But the rechercheur deserves to be more highly regarded than he is. Villani encouraged all the students in the auditorium who were interested in the sciences to pursue careers in them . Some students, he acknowledged, were "allergic to science". That was all right too.
Cartier then began the real business of the afternoon: explaining the importance of Galois' work to students who may not yet have finished high school algebra.
Galois worked in the theory of polynomial equations of one variable. Cartier explained what it means to find a solution in terms of radicals. The problem of finding solutions in radicals for polynomial of different degrees is very ancient. From the Aristotelian age to the 15th century it preoccupied the Arabs, the Hindus and the Greeks.
A passionate interest in the polynomial of the 5th degree (the quintic) and related issues arose in 16th century Italy : the names of Tartaglia, Cardano and Ferraro are the most familiar.
Galois' real accomplishment for mathematicians does not come from his discovery that the quintic is not always solvable by radicals; this had been done before by Abel and was latent in the work of LaGrange, Cauchy, Fubini and others. His fame stems from his invention of methods that transformed the whole subject of algebra.
Cartier quote: "In science it is usually much more difficult to prove that something doesn't exist, than that it does."
Galois brought a new point of view: making calculations with symbols quite apart from their numerical value. Using the equation x² = 2 as an example, Cartier tried to convey the notion of 'ambiguity' or 'symmetry' to the roots of an equation. Cartier's generalizations of the properties of ± √ 2 to expressions in a letter D, without reference to any numerical argument under the square root sign, was excellent.
From this he headed directly to the topic of imaginary numbers. The imaginaries I and ñI were referred to as twins:"jumbles" . This was the lead in to the topic of general complex numbers and the "twin" relationship of a pair of complex conjugates. The essence of algebra, he stressed, was that one could do calculations with symbols alone, without having to translate them (either actually or potentially) into numbers.
Two situations unrelated to pure mathematics were invoked to illustrate the concepts of 'ambiguity', 'dualism', and 'symmetry': right- and left- handedness, and parity violations in particle physics. When Carl Sagan and the engineers at NASA were designing a rocket that would travel into outer space to let the rest of the universe know that we exist, they were faced with a dilemma: how does one convey the notions of right-handedness and left handedness? There are, seemingly, no intrinsic differences between the notions of "left" and "right". The solution they came up was based on Louis Pasteur's discovery that the "right" and "left" isomers of certain chemicals have different properties. Cartier illustrated parity violations in physics by drawings on the blackboard.
After reviewing the properties of complex conjugates Cartier made a brief sally into quaternions.
By this time Cartier had lectured for over an hour and I sensed that many of the students were tired. He went on for another half hour, by which time several students in the audience had fallen asleep. Yet his audience was very polite. When he finished they gave him a (customarily vociferous) French applause.
We tried to figure out how the payment could be made. I have an HSBC account in Paris and the Fondation representative thought there should be no problems involved with depositing a check of about 32 euros (I forget the exact amount) into my account.
A few months later, having exhausted all efforts to cope with the customary obstacles to logic and common sense in dealing with banks in France, they simply sent me a check for the amount, and I deposited it a year later when I made my usual visit to Paris in autumn.
Shortly before the opening of the Galois bicentenary conference, I learned that the Fondation Cartier would be soon opening the exhibition "Un Depaysement Soudain" to be held from October 21st to March 18th .( This phrase from Recoltes et Semailles, which I translated as "a sudden change of scenery", was re-translated by someone at the Fondation,for this exhibition, as "A Beautiful Elsewhere"Had they asked me, I would have suggested "Shangri-La"!).
The Fondation Cartier also sent dozens of invitation cards to the Institute Poincaré , inviting anyone who was interested to attend a free pre-opening on October 20th, together with journalists from the Parisian media, and some of the artists whose works were on display.
After visiting the exhibition, I sent this E-Mail letter to the media contact official at the Fondation Cartier, M. Simmonet. Here is an edited version of that letter:
Dear Mr. Simmonet: On the afternoon of October 20th, I visited the new exhibition at the Fondation Cartier:"Un Depaysement Soudain" in my capacity both as a mathematician and journalist . I arrived there at about 1:30 and left after an hour.
Unhappily, I was very disappointed by the exhibition. It is indeed very difficult to convey the essence of mathematics, modern mathematics in particular to a public audience. However, the clientele that visits the Fondation Cartier are a unique selection from the general public, namely those persons interested in contemporary art. In my opinion the exhibition fails in both ways: in terms of its relationship to mathematics, and in its artistic merits.
This does not mean that I didn't find some good things in the exhibition. The most successful display for me was that of the Ergo-Robots, an installation of electro-mechanical servo- mechanisms that can "learn" patterns of behavior from external inputs such as the hand-wavings of people watching them. The eerie white skulls designed by David Lynch were very funny . This display worked for me because there was someone on hand to explain how the interactions with the skull-bearing robots were programmed. He also encouraged me to "teach them electronically " by waving my hands over the display. This kind of audience participation is very important for getting across the ideas of modern science.
The films being shown on the ground floor are short documentaries, interviews with or lectures by celebrated local mathematicians, Cedric Villani, Mikhail Gromov, Alain Connes and others. They varied from good pedagogical content (Villani speaking about non-Euclidean geometry at a blackboard), to rather funny (Mikail Gromov on Darwinism and mathematical minds), to scarcely comprehensible by a lay audience (Connes) .
Herein lies one of the failings of the Fondation Cartier exhibition, one that it shares with so much that one sees in science museums: shows, installations, lectures, exhibitions about mathematics designed for the public should not be done by research mathematicians! Modern mathematics research with its dozens of branches has become a "Tower of Babel", even to the mathematicians themselves! Research mathematicians are too absorbed in their specialties, and not always endowed with pedagogical talent or skill in conveying ideas to the public. This task is best left to mathematics teachers, or to the rare researcher (and they certainly exist) with a talent for conveying abstract notions to a general audience.
Furthermore, your exhibition overlooks major areas of modern mathematics. Among the major areas ignored by this exhibition one can include:
(1) Logic, and the Foundations of Mathematics: It is conceded that it is not altogether obvious how the ideas of modern logic can be employed by visual artists. However Magritte did a very good job with his suspended pipe and message!
(2) Modern Algebra. It is very easy to make this field interesting to the public , and to artists in every medium and genre. I am surprised that an exhibition which brings together mathematicians and artists does not have some space devoted to group theory, the science of symmetry, a subject central to all the arts. The display that comes closest to this is the big hemisphere in which David Lynch has constructed kaleidoscopic images of "Penrose tilings".They were discovered by the famous cosmologist , Robert Penrose,during his collaborations with M.C. Escher. Lynch's images do have the capacity to create mass hypnosis; but there is no explanation anywhere about why these "Penrose tilings" are so fascinating to mathematicians, by virtue of their property of exemplifying the ultimate "a-symmetry" possible in nature, that is to say, non-computability!!
(3)History of Mathematics. One does find a "historical tableau" in the shape of a large blackboard on the ground floor. This tableau is both irrelevant and confusing (the ultimate criticism I make on all mathematics shows in science museums).The grand lines of the development of mathematical thought from the 19th century to the present are either poorly portrayed or absent altogether. The "dates" and "arrows" on this tableau are meaningless.
(4) Catastrophe Theory. Many artists have created works using the ideas in this field: one can start with Salvador Dali!
(5) Fractals and Chaos. The general public loves fractal designs, Mandelbrot and Julia sets, seething forms emerging from the seas of pure chaos..
(6) Knot Theory . Knots, links and braids, ought to be a natural choice for visual artists.
This is a partial list of major fields in mathematics, all very active that are passed over completely by the installations and displays of "Un Depaysement Soudain"
Needless to say, but sadly, there is some music but no poetry in the exhibition. Oulipo?
The worst exhibit of " Un Depaysement Soudain" is the installation constructed by David Lynch, Patti Smith and Mikhail Gromov. It has a constant, loud, electronic background. The noises are horrible. At most they reflect, I'm sorry to sad, the horror that the general public feels towards mathematics in general! Every now and then the cacophony is interrupted by the singing Of Patti Smith . The decibel level of this song is painful; everyone entering the installation complained about it. The song in itself conveys nothing; one can't even make out the words, (written by Gromov).
A film is projected on one wall of this installation. It consists of a grim parade of the covers of books by famous 18th and 19th century mathematicians, accompanied by ponderous quotes presented in both French and English. How do these things reflect the symbiosis of art and mathematics?
On the opposite wall a computer image is projected of some sort of abstract flame or boat or storm. It relates neither with the erudite books on the screen, nor the electronic horror music, nor the Patti Smith song.
The volume of the electronic music was so loud that even people standing outside the installation were unable to carry on normal conversations. For example, David Lynch himself was there on the day I showed up, giving a press conference to a dozen journalists. They had to shout to make themselves heard. Finally everyone had to move to another part of the exhibition.
The small number of things that I found of value in this exhibition were: a beautiful sculpture of a tratrix (a fragment of a hyperbolic plane in 3-Space); the ergo-robot installation ; and a few remarks from the film documentaries portraying mathematicians speaking about the nature of mathematical thought.
In terms of communicating of the "joy and wonder" of mathematics (yes, they really exist), or the "creative synthesis of mathematics and art", I saw very little that would correspond to these goals . And yet I know for a certainty that it is possible to do a great many things with them.
Sincerely Yours, Dr. Roy Lisker.
I happened to mention that the composition of the attendance at the Galois conference was at least 90% male,the rest women. On the other hand the administration of the IHP is 90% women with a few men. Karine said that she had the perfect anecdote to capture the essence of this situation:
15 years ago she spent some time at the Wissenschaftskolleg, an intellectuals think-tank in Berlin. She'd been told a story to the effect that the director of the institution had complained that too many women and young scholars were there at the time . This made her very angry and she wrote a letter to the director asking him if he had indeed made such a comment.
He did not reply directly to her inquiry;rather he deflected the implied accusation in an odd way. Ms. Chelmna ,he wrote, apparently you didn't realize that the most of the positions in administration of the Wissenschaftskolleg are filled by women!
What he meant was this: the director and all the sub-directors are men; the women to whom he was referring were the secretaries and the domestic staff!
(2) The conference banquet of the Galois bi-centenary conference was held on the second floor of the main building of the Ecole Normale Superièure . A professional catering outfit had prepared the snacks in a very unusual way: morsels of many different kinds, meat, vegetables, candies, cakes, were stuck upon toothpicks. then so dressed and disguised that one would think that all of them were pieces of candy. Putting them into one's mouth they turn out to be turkey, ham, cheese, carrots, salmon..
The party went on in full swing for an hour or so before everyone was called to attention to listen to the speeches. There were quite a number of them:the ENS has some hard work hard to do making its apologies to Evariste Galois!
Then it was announced that a piano recital would begin shortly in another part of the building. About 50 of us made our way over to a small reception room, bridge chairs on the floor and level with the piano, dark yet comforting illumination, windows shrouded by heavy curtains. The pianist, an ENS student, proposed to "improvise" for us on themes relating to Evariste Galois. He began with some playing around with the letters of "Evariste" , based on a somewhat arbitrary code connecting letters to notes (like the B.A.C.H. motif used by Schumann and others). In this way he came up with a few motifs suitable for improvisation . His harmonic language was a form of simplified Debussy/Ravel ; not too exciting, yet it did show that he'd learned his keyboard harmony.
When he'd finished he asked the audience for ideas. I spoke up: did he know, I asked, that Erik Satie had written 4 "Galoisiennes"?
Galoisiennes? What the hell are those? No one got the point, and I thought at first that my trial balloon had been punctured. Then someone said, "Oh! You mean Gnossiennes!"
"Yes. You know: Gnossiennes? Galoisiennes! Much the same!"
Laughter rippled around the room. There were enough people present who knew about Satie's"Gnossiennes"
"So",I asked the young pianist,"Can you invent a Galoisienne for us"?
Now he understood what I wanted. He returned to the piano and came up with a nice mixture of a standard Gymnopedie with echos of the slow movement of Ravel's G major piano concerto!
At the IHP the next morning I was complimented by several people for my discovery of Satie's Galoisiennes !