Evariste Galois (edited)

Here is the edited version version of my paper. I corrected a lot of the errors as well as simplified the mathematics for those of you who are slower.

One of my assignments this week was to write a short paper about a deceased mathematician who contributed in the field of abstract algebra. The mathematician of my choice was Evariste Galois (pronounced "Gal-wa"). Some of you may not have heard of him, but the most famous of his discoveries is to prove using group theory that there is no formula in which polynomials of degree 5 and above can be solved as opposed to quadratic formulas that most of us are familiar with. There are formulas for polynomials of degree 4 and 5 and these can be easily researched in some advanced mathematics book if you decide to explore them. Anyway, here is my short paper on Galois. Hope you enjoy it.

The Life and Death of Evariste Galois

We know that Abel’s death was caused by poverty, but Galois’ death was a consequence of the stupidity of others. “Galois’ short life serves as a great example of the triumph of crass stupidity over untamable genius. Throughout his life, his magnificent powers were shattered before the mass stupidity aligned against him, and he was beaten by one unconquerable fool after another.” [1]

Galois was born on October 25, 1811 and lived a happy life until he was twelve. The only teacher he had until then was his mother. She taught him Greek, Latin and religion where she imparted her own skepticism about religion to her son. Galois' father was an important man in the community and in 1815 he was elected mayor of Bourg-la-Reine. [2]

At the age of twelve, Galois started formal schooling at the Louis-le-Grand in Paris. The school was like a prison. The intolerable conditions led to a strike among the students, leaving those involved expelled. Galois was not one of them. Because of his mother’s excellent instructions, he won many prizes. He also gained something more lasting than prizes. He gained the “conviction that neither fear nor the utmost severity of discipline can extinguish, that is the sense of justice and fair dealing in young minds experiencing their first unselfish devotion. [1]

Even though Galois’ school experiences were tumultuous, it was his first exposure to mathematics. February 1827 was a turning point in Galois' life. He discovered the splendid geometry of Legendre and was greatly inspired. Later on, he read Abel’s works and absorbed the masterpieces of algebraic analysis at the mere age of fifteen [1]. He quickly became absorbed in mathematics and his director of studies wrote

It is the passion for mathematics which dominates him. I think it would be best for him if his parents would allow him to study nothing but this. He is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.

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 criticized for being original. [2]

In 1828 Galois took the examination of the École Polytechnique but failed because he did not know some basic mathematics [3]. Commenting on his failure, Terquem (editor of the Nouvelles Annales de Mathematiques) remarks

A candidate of superior intelligence is lost with an examiner of inferior intelligence. Because they don’t understand me, I am barbarian [1].

In 1828, at seventeen, Galois met a man who could understand his genius, Louis Richard. Richard recognized Galois’ talents and proclaimed that Galois should be admitted to the Polytechnique without examination. Richard also said that “This pupil has a marked superiority above all his fellow students; he works only at the most advanced parts of mathematics.”

At eighteen, Galois wrote his important research on the theory of equations and submitted it to the French Academy of Sciences for publication. The paper was given to Cauchy for refereeing. Cauchy promised to present this, but he forgot. Worse still, he lost Galois’ abstract. That was the last Galois has ever heard of Cauchy’s generous promise [1].

He failed the École Polytechnique examination again in 1829 shortly after his father’s suicide. The priest of Bourg-la-Reine forged Mayor Galois' name on malicious forged epigrams directed at Galois' own relatives. Galois' father was a good natured man and the scandal that ensued was more than he could stand. He hanged himself in his Paris apartment only a few steps from Louis-le-Grand where his son was studying [2].

His disturbed mental state after his father’s death was part of the reason of his failure. He also did mathematics almost entirely in his head and this annoyed his examiners [3]. This examination has become a legend. During the oral part of the examination, one of the inquisitors ventured to argue a mathematical difficulty with Galois. The man was both wrong and obstinate. Galois lost his patience. He knew he had officially failed. In his rage, he hurled the eraser at his tormentor’s face and hit him.

Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Férussac of a posthumous article by Abel which overlapped with a part of his work. Galois then took Cauchy's advice and submitted a new article On the condition that an equation is soluble by radicals in February 1830. The paper was sent to Fourier, the secretary of the Paris Academy, to be considered for the Grand Prize in mathematics. Fourier died in April 1830 and Galois' paper was never subsequently found and so never considered for the prize [2]. After these series of misfortunes, Galois exclaimed, “Genius is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity” [1].

Galois then spent most of the last year and a half of his life in prison for revolutionary political offenses. To make matters worse, he received a rejection of his memoir in prison. Poisson had reported that “His argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor”. In March 1832 a cholera epidemic swept Paris and prisoners, including Galois, were transferred to the pension Sieur Faultrier. There he apparently fell in love with Stephanie-Felice du Motel, the daughter of the resident physician. After he was released on 29 April Galois exchanged letters with Stephanie, and it is clear that she tried to distance herself from the affair. The name Stephanie appears several times as a marginal note in one of Galois' manuscripts.

Galois fought a duel with Perscheux d'Herbinville on 30 May because of his political beliefs [1]. The night before the duel, Galois 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 mind before his death. He wrote in desperation and time and again his thoughts broke off and he wrote in the margin:

There is something to complete in this demonstration. I do not have the time.

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 1832 before his 21st birthday. His last wish was for Chevalier, his dearest friend, to send his papers to Gauss and Jacobi. 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.” [2]. His only enduring monument after his death is his collected works that filled sixty pages [1].

Significance of his work:

During the Middle Ages, mathematicians believed that unlike the quadratic equations, the algebraic formula for solving cubic equations was not possible. In the sixteenth century, Cardano, an Italian mathematician demonstrated that both cubic and quartic equations were solvable by radicals. With Cardano’s publications, mathematicians began in search for algebraic solutions for quintic polynomials. Abel built upon Lagranges’ work and managed to prove that for the general polynomial equation of degree 5 and above is not solvable by radicals. We know this from our ERES reading for Problem Set 4.

Galois extended the question of solvability by radicals in his Memoir on the Conditions for Solvability of Equations by Radicals. The memoir not only provided a theoretical framework for answering the solvability question, but also developed a framework for mathematical theory with far-reaching applications. He translated the problem of solvability into the language of field theory using a primitive form of the idea of an extension field [5]. Galois introduced the idea of what we refer to today as the Galois group. Using the Galois group, Galois was able to analyze a particular splitting field for a given polynomial. His results completely characterized the situation: a polynomial is solvable by radicals precisely when its Galois group admits a certain sequence of normal subgroups.

In solving this problem, Galois founded abstract algebra and group theory, which are fundamental to computer science, physics, coding theory and cryptography. Today, a "Galois connection" is a way of solving challenging mathematical problems by translating them into different mathematical domains, making the original problem amenable to a number of new solution techniques.

*Note to reader: For an example of a quintic polynomial that is not solvable by radicals, refer to page 559 in our textbook [3].

Conclusion:

Like Abel, Evariste Galois did not have a chance to live a full life and to witness the success of his works. Through a series of unfortunate events and wild coincidences, his talent was not duly noted. Cauchy had lost Galois’ abstract while Fourier passed away before he could judge his paper for a prestigious award. Today, Galois is not only remembered by his works, but also by the legends that were created about him. One can hardly forget the image a young talented mathematician throwing a chalkboard eraser at an examiner because of the examiner’s stubbornness and stupidity. Another legend about Galois was that he tried to write down everything he knew about group theory during the night before his duel which caused his tragic passing. To sum his misfortunes accurately, I repeat the quote from Galois:

“Genius is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity” [1].

Bibliography

[1] Bell, Eric Temple, Men of Mathematics, Simon and Schuster, 1937

[2] The MacTutor History of Mathematics archive,

http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Galois.html

[3] Gallian, Joseph A., Contemporary Abstract Algebra, Houghton Mifflin Company, 2002

[4] Wikipedia, the free encyclopedia, http://en.wikipedia.org/wiki/Galois

[5] Gellasch, Amy and Jardine, Dick, From Calculus to Computers, the Mathematical Association of America, 2005