悲劇數學天才Evariste Galois (1811-1832)

math618 · 发表于 2004-11-10 05:48:45

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><a href="http://www.npttc.edu.tw/academic/dpg/%E6%AF%8F%E6%9C%88%E7%A7%91%E5%AD%B8%E5%AE%B6%E4%BB%8B%E7%B4%B9/%E7%89%A9%E7%90%86%E5%B0%8D%E7%A8%B1%E4%B9%8B%E7%BE%8E%E7%9A%84%E6%95%B8%E5%AD%B8%E5%A5%A0%E5%9F%BA%E4%BA%BA-%E6%82%B2%E5%8A%87%E6%95%B8%E5%AD%B8%E5%A4%A9%E6%89%8DEvariste%20Galois(1811-1832).htm" target="_blank" >悲劇數學天才Evariste Galois (1811-1832) </A>

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align=center>Evariste Galois Childhood
Evariste Galois was born on the 25th of October 1811 in a small place called Bourg-la-Reine. This place is situated about 10 kilometres south of Paris. The Galois family accommodated a school, whose origins can be retraced to the times before the revolution. Prosperity and reputation of the family were based in this school.

His father Nicolas-Gabriel Galois his mother Adélaïde Marie Demante were very intelligent and were well trained in all the subjects considered important at that time: classical literature, religion and philosophy. However there is no sign of any mathematical ability in any of Galois' family. There is a record of Nicholas-Gabriel Galois gift for composing rhymed couplets. Evariste seemed to have inherited this talent. His cousin Mme. Bénard remembers decades later the family parties with their grandmother, when the children, i.e. Galois, his sister and their cousins recited rhymed couplets, which Evariste had composed.

At the age of ten his parents decided to send him to a college in Reims. But his mother seems to have changed her mind. She regarded her son as too small and defenceless to be sent off so far from home. He was allowed to stay home, enjoy the quiet family live and she kept on being his (as well as his sister's and brother's) sole source of education. His mother served as Galois' sole teacher until he was 12 years old. She taught him Greek, Latin and religion where she imparted her own skepticism to her son.

School Days
In 1823, at the age of twelve, Galois entered the lycée of Louis-le-Grand in Paris, which was his first school. Galois being a freshman was not yet involved in any political activity. According to his school report, Evariste did very well in the first school year and he received several prizes too. His successful start at school and the first two years show, that his mothers preparation was well founded, but during the 1825-26 school years his attitude to school changed. 

During the winter Galois was troubled by serious earache, which was caused by the bad conditions in the college buildings. During this time Galois’s mathematical genius was already stirring. In February 1827 Galois enrolled in his first mathematics class, a course by M. Vernier. It was during this course that Galois worked with Legendre's text on geometry and he was maybe for the first time acquainted with the theory of equations by Lagrange's works. His attitude towards the other subjects didn't change. His teachers complained that he didn't participate in lessons and hardly did any homework. But in the second school report of this class they wrote:

"Whats is dominating him is the fury of mathematics; also, I think that it would be better for him if his parents would agree to let him study solely mathematics. He is wasting his time here and he does nothing but torment his teachers and by doing so heaps punishments on himself."

Galois' school reports began to describe him as singular, bizarre, original and closed. It is interesting that perhaps the most original mathematician who ever lived should be criticised for being original. M. Vernier reported however 

Intelligence, marked progress but not enough method.


Strokes of Fate
In 1828 Galois took the examination of the École Polytechnique but failed. It was the leading University of Paris and Galois must have wished to enter it for academic reasons. However, he also wished to enter this school because of the strong political movements that existed among its students, since Galois followed his parents example in being an ardent republican. 

Back at Louis-le-Grand, Galois enrolled in the mathematics class of Louis Richard. However he worked more and more on his own researches and less and less on his schoolwork. He studied Legendre's Géométrie and the treatises of Lagrange. As Richard was to report 

This student works only in the highest realms of mathematics.

In April 1829 Galois had his first mathematics paper published on continued fractions in the Annales de mathématiques. On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Académie des Sciences. Cauchy was appointed as referee of Galois' paper.

Tragedy was to strike Galois for on 2 July 1829 his father committed suicide. His father hanged himself in his Paris apartment only a few steps from Louis-le-Grand where his son was studying. Galois was deeply affected by his father's death and it greatly influenced the direction his life was to take.

The two heavy strokes of fate, which overtook Galois in July 1829, were surely catalysts for the tragic course of his further life. Did they have a triggering effect or did they just accelerate what had started with his experiences at the Louis-le-Grand.

Just a few days after the unexpected death of his father, Galois took the the entrance examination to the École Polytechnique for the second time. It became a legend in the history of mathematics. He was aware that a refusal would be final this time, if he would flunk again. The examiners, though being recognized mathematicians, were not capable of detecting the mathematical genius of Evariste Galois. One of the two examiners asked the fatal question: He should describe the theory of the arithmetic logarithms. Galois criticized immediately the question, and mentioned to professor Dinet that there are no arithmetic logarithms. Why didn't he simply ask for the theory of the logarithms? Thereupon Galois refused to explain some propositions concerning logarithms. He said that it was completely obvious! He failed the examination.

According to the biographer Bell the course of events was different and more dramatic: Galois, used to unfold his mathematical thinking completely in his head, had a serious disadvantage before the board. The chalk and the sponge irritated him until he found a suitable application for one of the two things. One of the examiners had discussed both falsely and stubbornly a mathematical fact. In a fury and despair he hurled the sponge into the face of his tormentor. Twenty years later we find In the Nouvelles Annales Mathématiquesthis: "a candidate of superior intelligence was ruined by an examiner of minor intelligence. 


His Political Life
Galois must have been interested in politics when he was still attending the Louis-le-Grand. But his political activities started mainly at the Ecole Normale. The students of the Ecole Normale were set on fire by the events of the July revolution. Charles X had to flee from France and Louis-Philippe took over the crown. Cauchy, the famous mathematician, followed Charles into exile. 

Evariste Galois was expelled from school on the 9th of December - publicly announced on the 4th of January - because of an anonymous letter to the "Gazette des écoles". In this letter the director was blamed and offended for his behavior during the July revolution. Galois never admitted being the author of this letter neither to his fellow students nor to the director. In January 1831 Galois, no longer a student, and not a member of he National Guard anymore he tried to found a private algebra class to earn money for his living. His mother was not capable to support him financially. 

His course started as planned ont the 13th of January with about 40 students attending it, most probably mainly friends and not mathematicians. So it's no wonder that they couldn't follow his abstract lectures. Poisson asked Galois for a new copy of his memoire, so he wrote a new introduction and submitted his work on the 16th of January to the Academy.

　
On the 15th of June 15 he was tried for threatening the King's life. After a long trial the verdict was not guilty. A short while after the trial, Galois's memoir on the resolution of equations was rejected by the Academy. Poisson was a referee.

　

Fallen In Love
In the spring of 1832 Galois prison term came to an untimely and unexpected end. One of the worst world-wide raging cholera epidemics had arrived in France and was taking the lives of many people all over France. The prisoners in Sainte-Pélagie were in jeopardy, so the officials decided to transfer the youngest and those in bad health to a clinic. In this clinic, called after the owner Faultrier, he met Stéphanie, the daughter of Jean-Louis Poterin-Dumotel, one of the doctors. It's hard to recover the kind of relationship the two have been in. But from what is left of two letters, which he had received of her, we can be sure, that he was desparately in love with her. We don't know, if there has been a love affair or if she had turned him down immediately, but it looks as if her refusal was one of the main reason for his tragic death. 

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Fig. 1: The name Stephanie appears several times as <A>a marginal note</A> in one of Galois' manuscripts


The Duel
Galois fought a duel with Perscheux d'Herbinville on 30 May, the reason for the duel not being clear but certainly linked with Stephanie. 

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Fig. 2: The note reads: There is something to complete in this demonstration. I do not have the time.

It is this which has led to the legend that he spent his last night writing out all he knew about group theory. This story appears to have been exaggerated.

Galois was wounded in the duel and was abandoned by d'Herbinville and his own seconds and found by a peasant. He died in Cochin hospital on 31 May and his funeral was held on 2 June. It was the focus for a Republican rally and riots followed which lasted for several days. 

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Fig. 3: Galois’ Manuscipt

Galois' brother and his friend Chevalier copied his mathematical papers and sent them to Gauss, Jacobi and others. It had been Galois' wish that Jacobi and Gauss should give their opinions on his work. No record exists of any comment these men made. However the papers reached Liouville who, in September 1843, announced to the Academy that he had found in Galois' papers a concise solution.

...as correct as it is deep of this lovely problem: Given an irreducible equation of prime degree, decide whether or not it is soluble by radicals. 

Liouville published these papers of Galois in his Journal in 1846. The theory that Galois outlined in these papers is now called Galois theory.

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