A radical life EVARISTE GALOIS The long road to Galois CHAPTER 1 Babylon How many miles to Babylon? Three score miles and ten. Can I get there by candlelight? Yes, and back again. If your heels are nimble and

light, You may get there by candlelight.[ Babylon was the capital of Babylonia, an ancient kingdom occupying the area of modern Iraq. Babylonian algebra B.M. Tablet 13901front (From: The Babylonian Quadratic Equation, by A.E. Berryman, Math. Gazette, 40 (1956), 185B.M. Tablet

192) 13901-back Time Passes Centuries and then millennia pass. In those years empires rose and fell. The Greeks invented mathematics as we know it, and in Alexandria produced the first scientific revolution. The Roman empire forgot almost all that the Greeks had done in math and science. Germanic tribes put an end to the Roman empire, Arabic tribes invaded Europe, and

the Ottoman empire began forming in the east. Time passes Wars, and wars, and wars. Christians against Arabs. Christians against Turks. Christians against Christians. The Roman empire crumbled, the Holy Roman Germanic empire appeared. Nations as we know them today started to form. And we approach the year 1500, and the

Renaissance; but I want to mention two events preceding it. Al-Khwarismi (~790-850) Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was born during the reign of the most famous of all Caliphs of the Arabic empire with capital in Baghdad: Harun al Rashid; the one mentioned in the 1001 Nights. He wrote a book that was to become very influential Hisab al-jabr w'al-muqabala in which he studies quadratic (and linear) equations. It seems that ``al-jabr means

``completion and refers to removing negative terms. It is the origin of the word ``algebra. ``al-muqabala means balancing, and it refers to reducing positive terms if they appear on both sides of the equation. Al-Khwarismi (~790-850) Al-Khwarismi divides equations into six groups, then shows how to solve equations in each group. 1. Squares equal to roots. 2. Squares equal to numbers. 3. Roots equal to numbers. 4. Squares and roots equal to numbers; e.g. x2

+ 10 x = 39. 5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x. 6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2. Life was hard before the invention of decent mathematical notation! Al-Khwarismi (~790-850) His book was widely read also by European mathematicians, and they began to talk of doing things in the Al-Khwarismi way or, as it came to be known, by an algorithm.

What about the cubic equation? The Babylonians had solved some simple cubic equations. But the first serious attempt was perhaps due to a mathematician who is as famous as a mathematician as he is as a poet. Omar Khayyam (1048-1131). A great Persian mathematician working in the Seljuk (Turkish) empire. Solved cubic equations numerically by intersection of conic sections. Stated that some of these equations could not be

solved using only straightedge and compass, a result proved some 750 years later. The following web pages discuss his method: Omar Khayyam and a Geometric Solution of the Cu bic Omar Khayyam and the Cubic Equation Important general fact For all of the mathematicians trying to solve cubic equations, negative numbers and 0 were mysterious not well understood concepts. For us, the cubic equation is x3 + ax2 + bx + c =0. For us there is little difference, if any, between the

equations x3 + ax = b and x3 = ax + b. For Scipione dal Ferro, Tartaglia et al, the difference was essential because they could only understand the equation if a, b were positive numbers. And, of course, the equation was never written in the form x3 + ax + b = 0; setting to 0 just didn't make any sense. As the year 1500 approaches Only second degree equations were known to be solvable by radicals. And then But first, a few word from your real

numbers sponsor. Solving an equation by radicals There are, of course many ways of solving equations. Once one knows there are roots, it is only a matter of time to find them. At the heart of what it means to solve an equation is the question of the meaning of numbers.

What is a number? The question is (sort of) easy to answer if the number is an integer, or even a rational number. But what really is the square root of 2. The number ? If Greek mathematicians had had digital computers and had developed fully the atomic theory of matter, there

might not have been a need for irrational numbers. Chances are nature is discrete, and we only needed irrational numbers because we could not deal with zillions of particles at once; we had to come up with a continuous model and invent calculus to be able to handle it. In the future, we might be able to dispense with continuous models. Raffaelo Sanzio: The School of Athens ~ 1510

If Greek mathematicians had had digital computers This purely discrete mathematics will be to current mathematics what a music based on only two notes would be to Mozarts music. But Personally, I prefer to think that while our world is probably discrete, it is based on a continuous blueprint, and we want to study the blueprint more than the somewhat imperfect

construction based on it. Raffaelo Sanzio: The School of Athens ~ 1510 If Greek mathematicians had had digital computers But, lacking computers, and not quite sure about atoms, the Greeks invented a continuous mathematics and discovered irrational numbers. They were baffled.

Raffaelo Sanzio: The School of Athens ~ 1510 Eudoxus of Cnidus In one of the most brilliant ``tours de force in mathematics, Eudoxus (408 BCE-355 BCE) solved the problem of the irrationals. A semi modern interpretation is that in many ways it is meaningless to ask, for real numbers, is a = b? By Eudoxus, a =b simply means that both a < b and b < a are false. This idea lies behind the finding of formulas for areas and volumes of curved figures by Euclid and Archimedes; it is essential to the notion of convergence.

The point is That one has to be very explicit by what one means by having a formula to solve algebraic equations. To solve an equation by radicals means to have a formula in which the solutions are expressed as a function of the coefficients; the process of going from the coefficients of the equation to the solution should involve only the usual arithmetic operations (+, -, , /), and extraction of roots, and

The Italian Connection THE CUBIC CHRONICLES Luca Pacioli (1445-1509) In Summa de arithmetica, geometria, proportioni et proportionalit, published 1494 in Venice, he summarizes all that was known on equations. He discusses quartic equations stating that the equation that in modern notation is written as x4 = a + bx2

can be solved as a quadratic equation but x4 + ax2 = b and x4 +a = bx2 are impossible at the present state of science; ditto the cubic equation. Great friend of Leonardo da Vinci, briefly a colleague of Scipione dal Ferro. And Then.. THE FIRST ADVANCE OF EUROPEAN MATHEMATICS SINCE THE TIME OF THE GREEKS: Scipione dal Ferro figures out how to solve

the depressed cubic equation (ca. 1515) The depressed cubic equation 3 x px q Says cubi: I am depressed because I am missing my quadratic term. Scipione dal Ferro (14651526). There may not be any reliable portrait of Scipione dal Ferro on the web. One

that pops up is really Tartaglia. Professor at Bologna. Around 1515 figured out how to solve the equation x3 + px = q by radicals. Kept his work a complete secret until just before his death, then revealed it to his student Antonio Fior. Antonio Fior (1506-?). Very little information seems to be available about Fior. His main claim to fame seems to be his challenging Tartaglia to

a public equation ``solvathon, and losing the challenge. No portrait of Antonio Fior seems to be available. The two great rivals Nicolo of Brescia who adopted the name Tartaglia (Stutterer) (1499-1557). Hearing a rumor that cubic equations had been solved, figured out how to solve equations of the form x3 +mx2 = n, and made it public. This made Fior

think that Tartaglia would not know how to deal with the equations del Ferro knew how to solve and he challenged Tartaglia to a public duel. But Tartaglia figured out what to do with del Ferros equation and won the contest. Girolamo Cardano (1501-1576). Mathematician, physician, gambler (which led him to study probability), a genius and a celebrity in his day. Hearing of Tartaglias triumph over Fior convinced Tartaglia to reveal the secret of the cubic, swearing solemnly not to publish before Tartaglia had done so. Then he published first. But there are some possible excuses for this behavior.

Cardanos oath I swear to you, by God's holy Gospels, and as a true man of honour, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them. And then, in Ars Magna published in 1545, Cardano revealed the formula to the world. Completing the Picture Lodovico Ferrari (1522-1565). A protg of Cardano, he discovered how to solve the quartic equation, by reducing it to a

cubic. But his result could not be published before making public how to solve cubic equations. Tartaglia still not publishing, and the discovery that Scipione dal Ferro had already solved the cubic, where part of the reasons why Cardano published first. Not a picture of L. Ferrari As Ferrari wrote: Four years ago when Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal Della Nave, a clever and humane man who showed us a little

book in the hand of Scipione del Ferro, his father-in-law, written a long time ago, in which that discovery was elegantly and learnedly presented. The simple, but ingenious idea Tartaglias Solution of the Cubic Equation Quando chel cubo con le cose appresso Se agguaglia qualche numero discreto Trouan dui altri differenti in esso.

Dapoi terrai questo per consueto Che'llor produtto sempre sia eguale Alterzo cubo delle cose neto, Del numer farai due tal part' uolo Che l'una in l'altra si produca schietto El terzo cubo delle cose in stolo Delle qual poi, per communprecetto Torrai li lati cubi insieme gionti Et cotal somma sara il tuo

concetto. El residuo poi suo generale Delli lor lati cubi ben sottratti Varra la tua cosa principale. El terzo poi de questi nostri conti Se solue col secondo se ben guardi Che per natura son quasi congionti. In el secondo de cotestiatti Quando che'l cubo restasse lui

solo Tu osseruarai quest'altri Questi trouai, & non con pai tardi Nel mille cinquecent, quatroe trenta Con fondamenti ben sald' 01) When the cube with the cose beside it 02) Equates itself to some other whole number, 03) Find two other, of which it is the difference. Translated by: 04) Hereafter you will consider this customarily Friedrich Katscher . For more 05) That their product always will be equal

details, see: 06) To the third of the cube of the cose net. http://mathdl.maa.org/mathDL/ 07) Its general remainder then 46/?pa=content&sa=viewDocu 08) Of their cube sides, well subtracted, ment&nodeId=2433&pf=1 09) Will be the value of your principal unknown. 10) In the second of these acts, 11) When the cube remains solo 12) You will observe these other arrangements: The year was 1534, the ``city 13) Of the number you will quickly make two girded around by the sea is such parts,

Venice. 14) That the one times the other will produce straightforward 15) The third of the cube of the cose in a multitude, 16) Of which then, per common precept, 17) You will take the cube sides joined together. 18) And this sum will be your concept. 19) The third then of these our calculations 20) Solves itself with the second, if you look well after, 21) That by nature they are quasi conjoined. 22) I found these, & not with slow steps, 23) In thousand five hundred, four and thirty

In Symbols An important side-effect Rafael Bombelli (15261572) Engineer and self-taught mathematician, began writing in 1557 his masterwork, Algebra. It was to be five volumes long, but only three were ready for publication in 1572, the year he died. Made sense of the complex expressions that appeared as solutions to cubic equations. Because of that, MacTutor calls him the inventor of complex numbers. But it would take still quite

a while for these numbers to make true sense; all the way to the work of Euler some two centuries in the future. And now on to the QUINTIC!! And another couple of centuries pass. Very important developments during these centuries were: The development of mathematical notation,

much of it due to Franois Vite (15401603) with final touches by Ren Descartes (1596-1650) AND THEN: Calculus, a sea change! Nature and natures laws lay hid in night God said, `Let Newton be, and there was light! (Alexander Pope-1730) AND THE WORLD WAS NEVER THE SAME

The Bernoulli brothers, Jacob (1654-1705) and Johann (16671748) Leonhard Euler (17071783) What was the sea change? First of all, Newton did not invent Calculus out of nothing. Fermat, Descartes, Pascal, Barrow, among others got very close. Moreover, Leibniz also came up with similar ideas. I think the change that occurred in those years was that mathematics changed from being a static science to a dynamic one. The concept of function, called fluent by Newton, became a central concept in mathematics, perhaps the central concept. Newton did not invent calculus alone, nor did he invent all of it. The work of the Bernoulli brothers, among others, and, above

all, the work of Euler, one of the greatest mathematicians of all time, was almost as essential. And it would take another century before the concept of function was clearly understood. But back to equations. A very important development in this area was the statement and proof of the Fundamental Theorem of Algebra: Every algebraic equation of degree 1, with complex coefficients, has at least one complex root. The first proof of this theorem is

probably due to Jean Robert Argand (1768-1822); in his doctoral thesis Carl Friedrich Gauss (1777-1855) gave six different proofs. Gauss, in 1828 Someone who deserved more recognition. Paolo Ruffini (1765-1822) trained both as a mathematician and as a physician. A professor of the Foundations of Analysis in Modena (balsamic vinegar country), he was forced to resign his position and was barred from teaching because,

on religious grounds, he refused to swear an oath of allegiance to the Cisalpine Republic, an invention of Napoleon consisting of Lombardy, Bologna, Emilia, and Modena. He dedicated himself to the practice of medicine (he was licensed as a physician) and to mathematical research. He discovered that the quintic equation could not be solved by radicals. Someone who deserved more recognition. Ruffini was a great admirer of Lagrange, who had worked on trying to solve the quintic, without success. In the process Lagrange sowed the seeds

of what is now group theory, because he worked with permutations, but never composed them. Ruffini had to invent some of the notions of group theory. In 1799 he published a book with a rather descriptive title: Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto (General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible) Someone who deserved more recognition.

The book begins with: The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof. And nobody paid much attention, not even the immortal Lagrange. Most mathematicians could not really understand Ruffinis arguments, and did not believe them. Only Cauchy, who was influenced by Ruffinis ideas, in a reversal of his

usual role of stealing credit, gave Ruffini credit for an important discovery. Niels Henrik Abel (18021829) At the time of Abels birth, Norway was part of Denmark. When he was 12 years old, it became part of Sweden, eventually a semi-autonomous part. All this was the result of wars, partially the Napoleonic wars. Norway suffered mass starvation at one point; it was not a good time to be a Norwegian. Abel showed no special talent in mathematics until about age 15; then the school he attended

hired a good mathematics teacher, Berndt Holmbo, who saw Abel had talent. Niels Henrik Abel (18021829) Abels father was both prominent inpolitics, and a drunkard. He died in 1820 leaving the family without an income. Thanks to Holmbo who got him a scholarship, Abel finished his secondary studies and could enter the University of Christiania (Christiania being now Oslo). In 1822 he graduated and discovered, so he thought, how to solve the quintic equation by radicals. The editor of the journal to which he submitted the paper asked him for a numerical

example, and Abel discovered his mistake. Niels Henrik Abel (18021829) Abel published in 1824, at his own expense, a paper showing that the quintic was not solvable by radicals. He went on to do very important work in the theory of elliptic integrals, a theory he literally turned around because he inverted the problem, and came up with elliptic functions. He wrote paper after paper, always poor, not quite recognized, taking on odd jobs. While in Berlin he initiated a close friendship with August Leopold Crelle, who in 1826 began publishing a mathematical journal, Crelles Journal. It still

exists with the name Journal fr die reine und angewandte Mathematic. Its first volume contained six articles by Abel. Niels Henrik Abel (18021829) Crelle worked hard to get Abel a decent position in Berlin, and he finally succeeded. He wrote Abel the good news on April 8, 1829 with the good news. Unfortunately, Abel had died on April 5. In 1830 the Paris Academy of Sciences awarded Abel and Jacobi the Grand Prix for their work on elliptic functions. In January of 2002, the year of the second

centennial of Abels birth, the Norwegian government established the Abel prize, as a counterpart to the Nobel prize. Niels Henrik Abel (18021829) Quotes from Abel: My eyes have been opened in the most surprising manner. If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum had been rigorously determined. In other words, the most important parts of mathematics stand without foundation. It is true that most of it is valid, but that is very surprising. I struggle to find a reason

for it, an exceedingly interesting problem. (In a letter to Holmbo) Niels Henrik Abel (18021829) Quotes from Abel: Until now the theory of infinite series in general has been very badly grounded. One applies all the operations to infinite series as if they were finite; but is that permissible? I think not. Where is it demonstrated that one obtains the differential of an infinite series by taking the differential of each term? Nothing is easier than to give instances where this is not so.

Niels Henrik Abel (18021829) Quotes from Abel: The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes. Letter to Holmbo; quoted by G.H. Hardy at the beginning of Divergent Series. Niels Henrik Abel (18021829) Quotes from Abel: By studying the masters, not the pupils.

Supposedly he said this as reply to the question on how he got to be so good in mathematics. The only reference I have so far is Eric Temple Bell, which means that this is perhaps the most famous of quotes, and might not be a real quote at all. CHAPTER 2: THE SHORT AND TRAGIC LIFE OF EVARISTE GALOIS Some facts. Evariste Galois was born October 25,

1811. He was shot in a mysterious duel, May 30, 1832; died the next day. He was not quite 21 years old. The romantic version All night long he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death he saw could overtake him. Time after time he broke off to scribble in the margin "I have not time; I have not time," and passed on to the next

frantically scrawled outline. What he wrote in those last desperate hours before the dawn will keep generations of mathematicians busy for hundreds of years. He had found, once and for all, the true solution of a riddle which had tormented mathematicians for centuries: underwhat conditions can an equation be solved? (Eric Temple Bell, Men of Mathematics, Simon & Schuster, 1937) Further Facts: April 1829: Dmonstration dun Thorme sur les Publications Fractions Continues Priodiques, Annales de

Mathmatiques Pures et Appliqus. April 1830: Analyse dun Mmoire sur la Rsolution Algbrique des quations, Bulletin des Sciences Mathmatiques, Physiques et Chimiques. June 1830: Note sur la Rsolution des quations Numriques, Bulletin des Sciences Mathmatiques, Physiques et Chimiques.

June 1830: Sur la Thorie des Nombres, Bulletin des Sciences Mathmatiques, Physiques et Chimiques. December 1830: Notes sur Quelques Points dAnalyse, Annales de Mathmatiques Pures et Appliqus. January 1831: Letter to the editor of La Gazette des coles. And more: His most famous work, titled Mmoire sur les conditions de rsolubilit des quations par radicaux, now known as his First Memoir, was submitted to the

Academy of Sciences of Paris January 17, 1831. March 31, 1831 Mr. President: I dare to hope that Messrs Lacroix and Poisson will not take it badly that I recall to their memory a memoir relating to the theory of equations with which they were charged three months ago. The research contained in this memoir formed part of a work which I submitted last year in competition for the prize in mathematics, and in which I gave, for all cases, rules to recognize whether a given equation was or was not solvable by radicals. Since, until now, this problem had appeared to geometers to be, if not impossible, at least very difficult, the examining committee judged a priori that I could not

have solved this problem, in the first place because I was called Galois, and further because I was a student. And the committee lost my memoir. And someone told me that my memoir was lost. This lesson should have been enough for me. All the same, on the advice of an honorable member of the Academy, I reconstructed part of my memoir and presented it to you. You see Mr. President that my research has suffered up to now almost the same fate as that of the circle squarers. Will the analogy be pushed to its conclusion? Be so kind Mr. President as to relieve my disquiet by inviting Messrs. Lacroix and Poisson to declare whether they have lost my memoir or whether they have the intention to make a report of it to the Academy. The Verdict Poisson writing for both Lacroix and himself,

rejects the article on July 4, 1831. He mentions that there is some overlap with work of Abel, but it could perhaps go further, but it was very obscure, and unclear, and incomplete. One may therefore wait until the author will have published his work in its entirety before forming a final opinion; but given the present state of the part he has submitted to the Academy, we cannot propose to you that you give it your approval. A brief Chronology (1804- Napoleon becomes emperor of

France. ) 10/25/1811-Born in Bourg-la-Reine, a suburb of Paris. Second of three children, his sister Nathaly-Thodore was two years older, brother Alfred three years younger. (1812-Napoleon invades Russia with an army of a million men; only 10,000 return.) (1814-Napoleon is forced to abdicate, exiled to Elba. The brother of executed Louis XVI is installed as king of France, as Louis XVIII. ) A brief Chronology (1815- Napoleon escapes Elba, returns to

France. Louis XVIII and his court run away. But Napoleon is finally and decisively defeated in the battle of Waterloo. Louis XVIII returns as king of France.) 10/6/1823-Entered the Collge Louis-le-Grand. (1824-Louis XVIII dies, his younger brother becomes king as Charles X.) August 1828-Failed to be admitted to the prestigious cole Polytechnique. A brief Chronology 7/2/1829- Nicolas-Gabriel Galois, E.s father,

commits suicide. July or August 1829-Last attempt to be admitted to the cole Polytechnique. Fails again. November 1829: Entered the cole Preparatoire (better knwn now as cole Normale Suprieure.) (July 26-29, 1830. Workers, students, common folk pour onto the streets of Paris and set up barricades. It is a full revolution against Charles X. Charles X was deposed.) A brief Chronology Students from the Polytechnique were at the

forefront of the revolution, but the students of the cole Normale were not allowed on the streets by order of the director. Galois was furious! His letter to the Gazette des coles was a criticism of this director. January 1831- He is expelled from the cole Normale. (1830-A liberal minded nobleman, LouisPhilippe, is proclaimed king of France. He would be deposed in 1848.) A brief Chronology April 1831-Galois takes part in a celebration of the acquittal of some radical republicans in a

tavern in Paris. Alexandre Dumas was there and writes: Suddenly the name Louis-Philippe, followed by five or six whistles, catches my ear. I turned around. One of the most animated scenes was taking place fifteen or twenty seats from me. A young man, holding in the same hand a raised glass and an open dagger, was trying to make himself heard. He was variste Galois one of the most ardent republicans. All that I could perceive was that there was a threat, and that the name of Louis-Philippe had been pronounced; the intention was made clear by the open knife. A brief Chronology

Courageously, Dumas decides to escape by jumping out of an open window. 5/10/1831-Arrested for offensive political behavior (the tavern incident), but then acquitted and released. 7/14/1831 (Bastille Day)-He leads a crowd of 600 republicans across the pont Neuf; he was dressed in the uniform of the outlawed Artillery of the National Guard, and was carrying a loaded carbine, two pistols and a knife. He is arrested, held at Ste. Plagie prison. 10/23/1831-Convicted of carrying fire arms and wearing a banned uniform, sentenced to 6 more

months in Ste. Plagie. A brief Chronology Late May 1832-Engaged to duel. It is not really known with whom or why. 5/29/1832- Writes his Lettre Testamentaire, addressed to his friend Auguste Chevalier and revised some of his manuscripts. 5/30/1832- Shot in an early morning duel; died a day later in the Cchin hospital in Paris. 3/16/1832-Released from Ste. Plagie during an outbreak of cholera in Paris.