p. It also contains an incomplete study of two-dimensional linear groups over p. The Troisième Mémoire, which he described as being on the theory of integrals and ELLIPTIC FUNCTIONS [V.31], has never been found.b. Bourg-la-Reine, France, 1811; d. Paris, 1832
Theory of equations; theory of groups; Galois theory; finite fields
Galois studied at home until he was eleven years old, then entered the Collège Louis-le-Grand in Paris, where he stayed for six years. He had, and gave his teachers, a difficult time there, but excelled in mathematics, in which he read advanced work of LAGRANGE [VI.22], GAUSS [VI.26], and CAUCHY [VI.29] alongside standard texts of the time. He attempted the entrance examination for the École Polytechnique prematurely in June 1828, but failed. In July 1829, after his father’s suicide, Galois was again rejected by the École Polytechnique. He entered the École Préparatoire (later known as the École Normale Supérieure) in October 1829 but was expelled in December 1830 for unacceptable behaviour arising from political disagreements with the authorities. Arrested on Bastille Day (14 July) 1831, he spent the next eight months in prison for flouting authority again. He emerged at the end of April 1832 but somehow got himself challenged to a duel. On 29 May he edited his manuscripts and wrote a summary of his discoveries in a letter to his friend Auguste Chevalier. The duel took place the next morning and he died on 31 May 1832. Much has been written about him. But a man who dies so young leaves little real evidence for historians to work with, however rich his story, and most of his biographers have allowed romantic invention to colour their accounts of his life.
There are four main papers in Galois’s mathematical works (and a number of smaller and less important items). The first to be published was “Sur la théorie des nombres,” which appeared in April 1830 and contains the theory of Galois fields. These are analogues of the complex numbers obtained by adjoining to the integers modulo a prime number p a root of an irreducible polynomial congruence modulo p. The paper contains most of the basic features of what later became the theory of finite fields.
In the letter to Chevalier written on the eve of the duel, Galois mentions three memoirs. The first, now known as the Premier Mémoire, is a manuscript entitled “Sur les conditions de résolubilité des équations par radicaux.” Galois submitted work on the theory of equations to the Paris Academy on 25 May and 1 June 1829, but this is now lost and it seems quite possible that Galois withdrew it on the advice of Cauchy (to whom it had been given to referee) in January 1830. In February 1830 he resubmitted his work in competition for the Grand Prix de Mathématiques, but his manuscript was unfortunately and mysteriously lost on the death of FOURIER [VI.25] (and the prize was awarded jointly to ABEL [VI.33], posthumously, and JACOBI [VI.35]). Encouraged to do so by POISSON [VI.27] he submitted his ideas to the Academy for a third time in January 1831. It is this third submission (which was read by the Academy referees, Poisson and Lacroix, and rejected on 4 July 1831) that survives as the manuscript of the Premier Mémoire. This is the remarkable work in which he introduced what is now called the Galois group of an equation and showed how solubility of the equation in terms of radicals could be precisely characterized by a property of the group. It was the Premier Mémoire which turned the theory of equations into what is now called GALOIS THEORY [V.21].
The Second Mémoire also exists. Galois never completed it, however, nor is it all correct. Nevertheless, it is an exciting document that focuses on aspects of what is now recognized as the theory of groups. Its main theorem is (in group-theoretic language) that every primitive soluble permutation group has degree a power of a prime number and may be represented as a group of affine transformations over the prime field p. It also contains an incomplete study of two-dimensional linear groups over p. The Troisième Mémoire, which he described as being on the theory of integrals and ELLIPTIC FUNCTIONS [V.31], has never been found.
Galois’s main work—comprising the paper “Sur la théorie des nombres,” the Premier Mémoire, the Second Mémoire, and the letter to Chevalier—was finally published by LIOUVILLE [VI.39] in 1846. A critical edition by Bourgne and Azra, including every known fragment of Galois’s writing, was published in 1962.
Galois’s legacy is enormous. His ideas led directly to “abstract algebra” (see [II.3 §6]): when the abstract notion of field developed later in the nineteenth century, it turned out that most of the theory of finite fields had already been anticipated in that first paper; Galois theory developed directly out of the material in the Premier Mémoire; and the theory of groups developed from the ideas in the Premier Mémoire and the Second Mémoire together with a series of papers published by Cauchy in 1845.
Bourgne, R., and J.-P. Azra, eds. 1962. Écrits et Mémoires Mathématiques d’Évariste Galois. Paris: Gauthiers-Villars.
Edwards, H. M. 1984. Galois Theory. New York: Springer.
Taton, R. 1983. Évariste Galois and his contemporaries. Bulletin of the London Mathematical Society 15:107-18.
Toti Rigatelli, L. 1996. Évariste Galois 1811-1832, translated from the Italian by J. Denton. Basel: Birkhäuser.