GALOIS, VARISTE

born Oct. 25, 1811, Bourg-la-Reine, near Paris died May 31, 1832, Paris French mathematician famous for his contributions to the part of higher algebra known as group theory. His theory solved many long-standing unanswered questions, including the impossibility of trisecting the angle and squaring the circle. Galois was the son of Nicolas-Gabriel Galois, an important citizen in the Paris suburb of Bourg-la-Reine. In 1815, during the Hundred Days regime that followed Napoleon's escape from Elba, his father was elected mayor. Galois's mother, Adelade-Marie Demante, was of a distinguished family of jurists. She educated Galois at home until 1823, when he entered the Collge Royal de Louis-le-Grand. There his education languished at the hands of mediocre and uninspiring teachers. But his mathematical ability suddenly appeared when he was able to master quickly the works of Adrien-Marie Legendre on geometry and Joseph-Louis Lagrange on algebra. Under the guidance of Louis Richard, one of his teachers at Louis-le-Grand, Galois's further study of algebra soon led him to take up a major challenge. Mathematicians for a long time had used explicit formulas, involving only rational operations and extractions of roots, for the solution of equations up to degree four. (For example, 3x2 + 5 = 17 is an equation of the second degree, since it contains the exponent 2; solving an equation of this type is called a solution by radicals, because it involves extracting the square root of an expression composed of one or more terms whose coefficients appear in the equation.) The solution of quadratic, or second degree, equations goes back to ancient times. Formulas for the cubic and quartic were published in 1545 by Gerolamo Cardano, Italian mathematician and physician, after their discovery a few years earlier by the mathematicians Niccolo Tartaglia and Ludovico Ferrari. The equation of the fifth degree then defeated mathematicians until Paolo Ruffini in 1796 attempted to prove the impossibility of solving the general quintic equation by radicals. Ruffini's effort was not wholly successful, but the Norwegian mathematician Niels Abel in 1824 gave an essentially correct proof. Galois was unaware of Abel's work in the first stages of his investigation, although he did learn of it later. This was perhaps fortunate because Galois actually had launched himself on a much more ambitious study; while yet a student, at about age 16, he sought, by what is now called the Galois theory, a deeper understanding of the essential conditions that an equation must satisfy in order for it to be solvable by radicals. His method was to analyze the admissible permutations (a change in an ordered arrangement) of the roots of the equation. That is, in today's terminology, he formed the group of automorphisms (a particular kind of transformation) of the field, obtained by adjoining the roots of the equation. His key discovery, brilliant and highly imaginative, was that solvability by radicals is possible if and only if the group of automorphisms is solvable, which means essentially that the group can be broken down into prime-order constituents (prime numbers are positive numbers greater than 1 divisible only by themselves and 1) that always have an easily understood structure. The term solvable is used because of this connection with solvability by radicals. Thus Galois perceived that solving equations of the quintic and beyond required a wholly different kind of treatment than that required for the quadratic, cubic, and quartic. While still at Louis-le-Grand he published several minor papers. Soon disappointments and tragedy filled his life with bitterness. Three memoirs that he submitted to the Academy of Sciences were lost or rejected by the academicians, who as mathematicians were authorized to act as editors. The first was lost in 1829 by Augustin-Louis Cauchy. In each of two attempts (1827 and 1829) to enter the cole Polytechnique, the leading school of French mathematics, he had a disastrous encounter with an oral examiner and failed. Then his father, after bitter clashes with conservative elements in his hometown, committed suicide in 1829. The same year, realizing that his career possibilities as a professional mathematician had ended, Galois enrolled as a teacher candidate in the less prestigious cole Normale Suprieure and turned to political activism. But he continued his research. A second memoir, on algebraic functions, which he submitted in 1830 to the Academy of Sciences, was lost by Jean-Baptiste-Joseph Fourier. The revolution of 1830 sent the last Bourbon monarch, Charles X, into exile. But republicans were deeply disappointed when yet another king, Louis-Philippe, ascended the throneeven though he was a citizen king who wore the tricolour of the Revolution. When Galois wrote a vigorous article expressing these views, he was promptly expelled from the cole Normale Suprieure. Subsequently he was arrested twice for republican activities; he was acquitted the first time but spent six months in prison on the second charge. His third memoir in 1831 was returned by Simon-Denis Poisson with a note that it was virtually incomprehensible and should be expanded and clarified. The circumstances that led to Galois's death in a duel in Paris have never been fully explained. It has been variously suggested that it resulted from a quarrel over a woman, that he was challenged by royalists who detested his republican views, or that an agent provocateur of the police was involved. Alexandre Dumas, in his autobiography Mes Mmoirs (186365), implicated Pcheux d'Herbinville as the man who shot Galois. In any case, anticipating his death in the coming duel, Galois in feverish haste wrote a scientific last testament addressed to his friend and former schoolmate Auguste Chevalier. In his distracted notes, there are hints that Galois had begun to develop the theory of algebraic functions, the full development of which was achieved 40 years later by the German mathematician Bernhard Riemann. Galois's manuscripts, with annotations by Joseph Liouville, were published in 1846 in the Journal de Mathmatiques Pures et Appliques. In 1870 the French mathematician Camille Jordan published the full-length treatment of Galois's theory, Trait des Substitutions. These works rendered his discoveries fully accessible and his place secure in the history of mathematics. On June 13, 1909, a plaque was placed on Galois's modest birthplace at Bourg-la-Reine, and the mathematician Jules Tannery made an eloquent speech of dedication, which was published the same year in the Bulletin des Sciences Mathmatiques. Irving Kaplansky