Evariste Galois

Évariste Galois (Bourg-la-Reine, October 25, 1811 - Paris, May 31, 1832) was a French mathematician. While still a teenager, he was able to determine the necessary and sufficient condition for an algebraic equation to be solved by radicals. He provided a solution to an open problem through the new concept of a group of permutations.His work provided the fundamental basis for the theory that bears his name, a major branch of abstract algebra. He was the first to use the term "group" in a mathematical context.

Galois theory constitutes one of the mathematical bases of CDMA modulation used in communications and, especially, in satellite navigation systems, such as GPS, GLONASS and others.^{[citation needed]}

Biography

Évariste Galois was born in Bourg-la-Reine, a town just outside of Paris. His father was Nicolas-Gabriel Galois, director of the local school who would go on to be elected mayor of the commune at the head of the liberal party, a supporter of Napoleon. His mother, Adelaide-Marie, was a person of undoubted intellectual qualities, the daughter of a highly influential family of lawyers in Paris.

Until the age of twelve, Évariste was educated by his mother, along with his older sister Nathalie-Théodore, obtaining a solid background in Latin and Greek, as well as the classics. He was a highly intelligent boy, but although many consider him to have been a mathematical prodigy, it is not likely that during his early education the young man had any in-depth exposure to mathematics (other than elementary arithmetic) and neither is he known to have that there had been instances of special mathematical talent in his family.

His academic education began at the age of 12 when he entered the royal Lycée Louis-le-Grand in Paris, where Robespierre and Victor Hugo had studied. There he had his first dalliances with political overtones (a confrontation with the director of the boarding school) that resulted in the expulsion of several students, among whom he was not, but who forged an incipient rebellion against authority (especially an anti-ecclesiastical and anti-monarchical ideology). which he kept until his death). During the first two years at the Lycée Louis-le-Grand, Galois had a normal performance and even won some prizes in Greek and Latin. But in third grade, his rhetoric paper failed and he had to repeat the course. It was then that Galois came into contact with mathematics: he was 15 years old at the time. After getting into mathematics, he became interested in geography.

The math program at the high school didn't differ much from the rest; however, Galois found in him the intellectual pleasure he lacked. The course taught by Ms. Vernier awakened Galois's mathematical genius. After effortlessly assimilating the official text of the school and the manuals in use, Galois began with the most advanced texts of that time: he studied Legendre's geometry and Lagrange's algebra. Galois deepened considerably in the study of algebra, a subject that then still had many gaps and obscure questions. And so he came to know the number of unresolved problems that that discipline contained. Problems that came to occupy most of his study time. He began to neglect other subjects, attracting hostility from humanities professors. Even Vernier suggested the need to work more in other different disciplines.

However, Galois had a clear idea: he wanted to be a mathematician and he wanted to enter the École polytechnique. Thus, he decided to take the entrance exam one year in advance (1828). Lacking fundamental training in various aspects and without having received the usual preparatory course in mathematics, Évariste was rejected. Galois did not accept this initial rejection, and it increased his rebellion and his opposition to authority. However, he continued to make rapid progress in the study of mathematics during the second year taught at the Lycée Louis-le-Grand, in this case by Ms. Richard, who was able to see the qualities of the young man and requested that he be admitted to the École polytechnique. Although Richard's request was not honored, the dedication and drive Galois received from his teacher had remarkable results.

While still a student at Louis-le-Grand, Galois managed to publish his first paper (a proof of a theorem on periodic continued fractions) and soon after he found the key to solving a problem that had plagued mathematicians for years. more than a century (the conditions for solving polynomial equations by radicals). However, his most notable advances were those related to the development of a new theory whose applications far exceeded the limits of algebraic equations: group theory.

Fate was not going to bring him many more successes. A few days before taking the second (and final) entrance exam to the École polytechnique, Évariste's father took his own life. In this context, Galois introduced himself and, with his usual rebellious ways and contempt for authority, refused to follow the examiners' directions by refusing to justify his statements. And, naturally, he was definitively rejected.

Forced to consider the then less prestigious École normale, Galois sat the baccalaureate exams (required for admission), and this time he passed thanks to his exceptional mathematics grades. Galois was admitted to the École normale at about the same time that his revolutionary papers on group theory were being evaluated by the Academy of Sciences. However, his articles were never published during Galois's lifetime. Initially he sent it to Cauchy, who rejected it because his work had points in common with a recent article published by Abel. Galois reviewed it and resubmitted it to him, and this time Cauchy submitted it to the academy for his consideration; but Fourier, the life secretary of the same and in charge of the publication, died shortly after receiving it and his memory was misplaced. The prize was awarded ex æquo to Abel and Jacobi, and Évariste accused the academy of a farce to discredit him.

Despite the loss of memory sent to Fourier, Galois published three articles that same year in the Bulletin des sciences mathématiques, astronomiques, physiques et chimiques of the Baron de Férussac. These works present the foundations of Galois's theory and, although it was an unfinished work, they prove without a doubt that the young man had gone further than any other mathematician in the field of algebra related to the resolution of polynomial equations.

By then, Galois's life was beginning to be tinged with a marked political tinge. In July 1830, the republicans rose up and forced King Charles X into exile. However, the triumph of the republicans, among whom was the young Galois, was crushed by the accession to the throne of a new king: Louis Philippe of Orleans. Galois took an active part in republican demonstrations and societies. He was expelled for it from the École normale. In the spring of 1831, at just 19 years old, Galois was arrested and imprisoned for more than a month, accused of sedition, after a defiant toast on behalf of the king. He was initially acquitted, but he was arrested again for another seditious attitude in July, and this second time he spent eight months in prison.

During that year of 1831, Galois had finally rounded up the pending issues in his work and had submitted it for the consideration of Poisson, who recommended that he resubmit it to the Academy. Later that same year, Poisson himself recommended to the Academy that it reject his work, stating that "his arguments of his were neither clear enough nor sufficiently developed to enable them to judge his rigor". Poisson himself, despite his enormous mathematical prestige and his efforts, failed to understand the results presented to him by that memory. Galois received the rejection letter in prison.

A month before his death, on April 29, 1832, Galois was released from his imprisonment. The details leading up to his duel (allegedly over a skirt mess) are unclear. What remains for history is the night before the event. Évariste Galois was so convinced of the imminence of his death that he spent the entire night writing letters to his republican friends and composing what would become his mathematical testament. In these latter papers, he briefly described the implications of the work he had developed in detail and annotated a copy of the manuscript that he had submitted to the academy along with other papers.

In the early morning of May 30, 1832, Galois lost a duel at pistols against the fencing champion of the French army, and died the next day at ten o'clock in the morning (probably of peritonitis), in Cochin hospital. His last words to his brother, Alfredo, were: "Don't cry! I need all my courage to die at twenty.

Galois's mathematical contributions were finally published in 1843, when Joseph Liouville revised his manuscripts. He declared that this young man, in fact, had solved Abel's problem by other means that represented a true revolution in the theory of mathematics used. The manuscript appeared in the October 1846 issue of the Journal des mathématiques pures et appliquées.

Mathematician's Day is celebrated in Argentina on May 31 in his memory.

Eponymy

• The lunar crater Galois bears this name in his memory.