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Memory

Algebraic obsessions

Two hundred years ago Évariste Galois, the creator of group theory and a tragic character in the history of science, was born

INSTITUT NATIONAL DES SCIENCES APPLIQUÉES (INSA) DE LYONSolving equations is important in order for one to reach a solution to problems that arise in day-to-day practices and situations, through the study of arithmetic and geometric issues and even as a pastime. They can be found in the earliest written records of mathematics from the ancient civilizations of Egypt, Babylon, India and China. Algebra as a method for solving equations only came into being in the eighth century in the Arab world, thanks to Mohammad al-Khwarizmi. From this time until the sixteenth century, mathematicians successfully focused on deciphering equations of the second, third and fourth degrees, by finding formulas with radicals for the solutions. However, they found themselves at a loss in the face of equations of the fifth degree or higher. Between 1826 and 1832, thanks to the work of Niels Abel (1802-1829) and Évariste Galois (1811-1832), it was shown that it is impossible to have a general formula with radicals for equations of degrees greater than five. Galois was born 200 years ago and left a contribution, group theory, which is considered one of the major intellectual achievements of the mathematical sciences. His tragic death at the age of 20 and the belated publication of his few papers resulted in his only becoming acknowledged in the second half of the nineteenth century.

Galois was born in Bourg-la-Reine, near Paris. At school, he was a highly irregular student, although he could read important mathematicians such as Joseph-Louis Lagrange, Adrien-Marie Legendre, Augustin-Louis Cauchy and Friedrich Gauss with great ease. There are records from his rhetoric teacher complaining that, at the age of 16, it was useless trying to get him to become interested in any discipline: “A slave to his passion for mathematics, he totally disregarded everything else.” It was this passion that led him to a major ambition: finding a means of solving fifth degree equations. He also wanted to get into the Polytechnic School, the country’s main institution of higher education. Twice he tried, unsuccessfully, to be accepted. According to those who have studied his life, he was probably ill prepared for this. One of the complaints of his examiners was that he worked out a substantial portion of the calculations in his head and only gave them the results, without showing how the problem had been solved. This made his examiners incredulous and displeased them. He then chose the Preparatory School, a temporary name assigned to the Teacher’s College.

The militancy of Galois in favor of the Republic in monarchist France led to his being expelled from the institution and being imprisoned twice. Having fallen in love with Stéphanie Potterin du Motel, he was killed in a duel, though it is not known by whom exactly. One version of the story indicates that the challenger was someone close to the girl. Another states that it was a machination of the monarchists. A third says that Galois himself provoked his death to fuel a rebellion against king Charles X. The only thing that is clear is that he was hit by a bullet in the belly and died on May 30 of 1832, after 12 hours.

Galois wrote five small articles and three memoirs. Overall, his mathematical works total 60 pages. During his lifetime, only these short articles were published. After his death, his mother gave several manuscripts and three of his letters to August Chevalier, one of his friends. Two of these were about politics, and one was a summary of his memoirs and was published in the magazine Revue Encyclopédique, in September of 1832. Only in 1846 were all his works published in Journal de Mathématiques Pures et Appliquées.

“Galois was a genius and created a veritable conceptual revolution,” says the mathematician Ubiratan D’Ambrosio, professor emeritus of the State University of Campinas (Unicamp), who studies the subject. “His greatest originality was devising pure abstraction. He considers a set of objects, with no reference to their nature, and defines a law of composition, similar to multiplication tables, for such objects. He talks about their properties and thus introduces the concept of a group,” he explains. Over time, this theory gave rise to concepts connected to abstract structures, such as bodies, rings and others. “A new algebra emerged from group theory,” states D’Ambrosio.

Marcos Teixeira, from Paulista State University (Unesp) in Rio Claro, who researches the history of mathematics, says that through the association of a group of permutations with an equation, Galois was able, by studying the properties of this group, to determine the impossibility of a general formula for the solution of equations of a degree equal to or greater than five. “This was revolutionary, but at the time of Galois, as is the case with all novel theories, things were unclear, so that it took some time for them to mature and be accepted.”

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