Évariste Galois

2008/9 Schools Wikipedia Selection. Related subjects: Mathematics

Évariste Galois
Image:Galois.jpeg Galois at the age of fifteen from the pencil of a classmate. He was young-looking for his age and had black hair.
Born	October 25, 1811(1811-10-25) Bourg-la-Reine, France
Died	May 31, 1832 (aged 20) Paris, France
Nationality	French
Fields	mathematics
Known for	his works on the theory of equations and Abelian integrals

Évariste Galois (IPA:  [evaʁist ɡaˈlwa]; October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of abstract algebra, and the subfield of Galois connections. He was the first to use the word "group" (French: groupe) as a technical term in mathematics to represent a group of permutations. A radical Republican during the monarchy of Louis Philippe in France, he died from wounds suffered in a duel under murky circumstances at the age of twenty.

Early life

Galois was born on October 25, 1811, to Nicolas-Gabriel Galois and Adélaïde-Marie (born Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party, and became mayor of the village after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and she was for the first twelve years of her son's life responsible for his education. At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home. In October 1823, he entered the Lycée Louis-le-Grand, and despite some turmoil in the school at the beginning of the term (where about a hundred students were expelled), Galois managed to perform well for the first two years, obtaining the first prize in Latin. He soon became bored with his studies, and it was at this time, at the age of 14, that he began to take a serious interest in mathematics. He found a copy of Adrien Marie Legendre's Éléments de Géométrie, which it is said that he read "like a novel" and mastered at the first reading. At the age of 15, he was reading the original papers of Joseph Louis Lagrange and Niels Henrik Abel, work intended for professional mathematicians, and yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.

Budding mathematician

In 1828, he attempted the entrance exam to École Polytechnique, without the usual preparation in mathematics, and failed. In that same year, he entered the École préparatoire, a far inferior institution for mathematical studies at that time, where he did find some professors sympathetic to him. In the following year, Galois' first paper, on continued fractions was published, and while it was competent it held no suggestion of genius. Nevertheless, it was at around the same time that he began making fundamental discoveries in the theory of polynomial equations, and he submitted two papers on this topic to the Academy of Sciences. Augustin Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. In spite of many claims to the contrary, it appears that Cauchy had recognized the importance of Galois' work, and it is speculated that he refused publication of the papers because he preferred Galois combine the two manuscripts into a single, more comprehensive paper (see below).

On July 28, 1829, Galois' father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois took his second, and final attempt at entering Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, he threw the rag used to clean up chalk marks on the blackboard at the examiner's head. More plausible accounts state that Galois refused to justify his statements and answer the examiner's questions. The recent death of his father may have also influenced his behaviour.

Having been denied admission to Polytechnique, Galois took the Baccalaureate examinations in order to enter the Ecole Normale. He passed, receiving his degree on December 29 1829. His examiner in mathematics reported: "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

His memoir on equation theory would be submitted several times but was never published in his lifetime, due to various events. As previously mentioned, his first attempt was refused by Cauchy, but he tried again in February 1830 after apparently following Cauchy's suggestions, and submitted it to the Academy's secretary Fourier, to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to Abel posthumously and also to Jacobi. Despite the lost memoir, Galois published three papers that year which laid the foundations for Galois theory.

Political firebrand

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with abdication, staged a coup d'état, and issued his notorious July Ordinances, touching off the July Revolution which ended with Louis-Philippe becoming king. While their counterparts at Polytechnique were making history in the streets during the les Trois Glorieuses, Galois and all the other students at the École Normale were locked in by the school's director. Galois was incensed and he wrote a blistering letter criticizing the director which he submitted to the Gazette des Écoles, signing the letter with his full name. Despite the fact that the Gazette's editor redacted the signature for publication, Galois was, predictably, expelled for it.

Even before his expulsion from Normale was to take effect on January 4, 1831, Galois joined the staunchly Republican Artillery unit of the National Guard. These and other political affiliations continually distracted him from mathematical work. Due to controversy surrounding the unit, soon after Galois became a member, on December 31, 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government.

In April, all nineteen officers were acquitted of all charges, and on May 9, 1831, a banquet was celebrated in their honour, with many illustrious personalities, such as Alexandre Dumas present. The proceedings became more riotous, and Galois proposed a toast to King Louis-Philippe with a dagger above his cup, which was interpreted as a threat against the king's life. He was arrested the following day, but was later acquitted on June 15.

On the following Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested, this time sentenced to six months in prison for illegally wearing a uniform. He was released on April 29, 1832. During his imprisonment, he continued developing his mathematical ideas.

Final days

Galois returned to mathematics after his expulsion from Normale, although he was constantly distracted in this by his political activities. After his expulsion from Normale was official in January 1831, he attempted to start a private class in advanced algebra which did manage to attract a fair bit of interest, but this waned as it seemed that his political activism had priority. Simeon Poisson asked him to submit his work on the theory of equations, which he submitted on January 17. Around July 4, Poisson declared Galois' work "incomprehensible", declaring that "[Galois'] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor", however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion." While Poisson's rejection report was made before Galois' Bastille Day arrest, it took some time for it to reach Galois, which it finally did in October that year, while he was imprisoned. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and he decided to forget about having the Academy publish his work, and instead publish his papers privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice and began collecting all his mathematical manuscripts while he was still in prison, and continued polishing his ideas until he was finally released in April 29, 1832.

A month after his release, on May 30, was Galois' fatal duel. The true motives behind this duel that ended his life will most likely remain forever obscure. There has been a lot of speculation, much of it spurious, as to the reasons behind it. What is known is that five days before his death he wrote a letter to Chevalier which clearly alludes to a broken love affair.

Some archival investigation on the original letters reveals that the woman he was in love with was apparently Mademoiselle Stéphanie-Felice du Motel, the daughter of one of the resident physicians in the prison where he was transferred shortly before his release and where he remained thereafter. Fragments of letters from her copied by Galois himself (with many portions either obliterated, such as her name, or deliberately omitted) are available, and they seem to indicate that he was rejected and took it badly. The letters also give some intimation that Mlle. du Motel had confided some of her troubles with Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by some of the other letters Galois later wrote to his friends the night before he died. Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' biographers (most notably by Eric Temple Bell in Men of Mathematics and by Fred Hoyle in Ten Faces of the Universe), such as the oft-repeated conjecture that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.

As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, one of the nineteen artillery officers on whose acquittal the banquet that occasioned Galois' first arrest was celebrated. However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent which is inconsistent with d'Herbinville, and more accurately describes one of Galois' Republican friends, most probably Ernest Duchatelet, who was also imprisoned with Galois on the same charges. Given the conflicting information available, the true identity of his killer may well be equally lost to history.

Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas. Hermann Weyl, one of the greatest mathematicians of the 20th century, said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. In these final papers he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the academy and other papers. On 30 May 1832, early in the morning, he was shot in the abdomen and died the following day at ten in the Cochin hospital (probably of peritonitis) after refusing the offices of a priest. He was 20 years old. His last words to his brother Alfred were:

Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans ! (Don't cry, Alfred! I need all my courage to die at twenty.)

Much of the drama surrounding the legend of his death has been attributed more to one source than any other, Eric Temple Bell's Men of Mathematics.

Galois' mathematical contributions were finally fully published in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal des mathématiques pures et appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not solvable by radicals. Note: Abel had already proved the impossibility of a "quintic formula" by radicals in 1824; Ruffini had published a solution in 1799 that turned out to be flawed. But Galois' methods led to deeper research in what is now called Galois Theory. For example, one can use it to determine, for any polynomial equation, whether or not it has a solution by radicals.