Évariste Galois

Évariste Galois (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 fundamental 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. He died in a duel at the age of twenty.

Early life

Galois was born on October 25, 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née 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 had for the first twelve years of her son's life been 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.

Budding mathematician

However, 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 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, 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, who, despite many claims to the contrary, appears to have recognized the importance of Galois' work, but refused to accept them for publication for reasons that still remain unclear, though it is speculated that he preferred Galois to 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 the 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 behavior.

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 the Galois theory.

Political firebrand

Galois grew into 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 gained a 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 the 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 the Normale was to take effect on January 4 of 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, in 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 honor. 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 the Normale, although he was constantly distracted in this by his political activities. After his expulsion from the 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 "^* 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, but it was supposedly fought to defend the honor of a girl with whom Galois had fallen in love. Other theories speculate that the duel was used as an excuse to eliminate a political enemy. 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 the 30th of 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 that he had indeed solved the problem first proposed and also solved by Abel. The manuscript was finally published in the October-November 1846 issue of the Journal des mathématiques pures et appliquées.

References

Laura Toti Rigatelli, Evariste Galois, Birkhauser, 1996, ISBN 3764354100. This biography challenges the common myth concerning Galois' duel and death.
Ian Stewart, Galois Theory, Chapman and Hall, 1973, ISBN 0412108003. This comprehensive text on Galois Theory includes a brief biography of Galois himself.

External links