Évariste Galois (25 October 1811 – 31 May 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 350 years-standing problem. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He died at age 20 from wounds suffered in a duel. Unsurprisingly, Galois's collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.
Galois was born to Nicolas-Gabriel Galois and Adélaïde-Marie. His mother was a fluent reader of Latin and classical literature and was responsible for her son's education for his first twelve years. 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, aged 12, he entered the Lycée Louis-le-Grand, where he managed to perform well for the first two years, obtaining the first prize in Latin. He soon became bored with his studies and, at the age of 14, 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 15, he was reading the original papers of Joseph Louis Lagrange, such as the landmark Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians.
In 1828, aged 17, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination.
In 1829, aged 18, Galois's first paper, on continued fractions, was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences.
Augustin Louis Cauchy refereed these papers, but refused to accept them, Cauchy recognized the importance of Galois's work, and suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time, considered Galois's work to be a likely winner. In February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Joseph Fourier, to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after, and the memoir was lost. He submitted his memoir on equation theory several times, but it was never published in his lifetime.
On 28 July 1829, Galois's father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior.
Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on 29 December 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."
In 1830, aged 19, Galois published three papers, one of which laid the foundations for Galois theory. The second one was about the numerical resolution of equations. The third was an important one in number theory, in which the concept of a finite field was first articulated.
Galois lived during a time of political turmoil in France. While their counterparts at the Polytechnique were making history in the streets during 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 wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name.
Although the Gazette's editor omitted the signature for publication, Galois was expelled. Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard.
On Bastille Day (14 July 1831), Galois was at the head of a protest, heavily armed with several pistols, a rifle, and a dagger. He was arrested. He was sentenced to six months in prison. Nine and a half months later, he was released, on 29 April 1832, aged 20. During his imprisonment, he continued developing his mathematical ideas.
Galois returned to mathematics after his expulsion from the École Normale; Siméon Poisson asked him to submit his work on the theory of equations, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois's work "incomprehensible", declaring that "[Galois's] 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 report was made before Galois's Bastille Day arrest, it took until October to reach Galois in prison. Galois reacted violently to the rejection letter, but did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832.
Galois's fatal duel took place on 30 May 1832, aged 20. The true motives behind the duel will most likely remain forever obscure but he wrote a letter to Chevalier which clearly alludes to an illicit love affair. His letters hint that a Mademoiselle Stéphanie-Félicie Poterin du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf. Given the conflicting information available, the true identity of his killer may well be lost to history, despite some claims that it was Mme Motel's fiance.
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, and three attached manuscripts. Mathematician Hermann Weyl 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."
Early in the morning of 30 May 1832, he was shot in the abdomen and died the following morning at ten o'clock in the Cochin hospital after refusing the offices of a priest. Galois was 20 years old. His last words to his younger 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.
Galois's mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de 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 generally solvable by radicals. Galois's 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 it has a solution by radicals.
Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes. Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.
Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.
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