Évariste Galois

French mathematician

• Born: October 25, 1811
• Birthplace: Bourg-la-Reine, near Paris, France
• Died: May 31, 1832
• Place of death: Paris, France

With the aid of group theory, Galois produced a definitive answer to the problem of the solvability of algebraic equations, a problem that had preoccupied mathematicians since the eighteenth century. Consequently, he laid one of the foundations of modern algebra.

Early Life

Èvariste Galois (gah-lwah) was the son of Nicolas-Gabriel Galois, a friendly and witty liberal thinker who headed a school that accommodated about sixty boarders. Elected mayor of Bourg-la-Reine during the Hundred Days after Napoleon’s escape from Elba, the elder Galois retained office under the second Restoration. Galois’s mother, Adelaïde-Marie Demante, was from a long line of jurists and had received a more traditional education. She had a headstrong and eccentric personality. Having taken control of her son’s early education, she attempted to implant in him, along with the elements of classical culture, strict religious principles as well as respect for a stoic morality. Influenced by his father’s imagination and liberalism, the eccentricity of his mother, and the affection of his elder sister Nathalie-Théodore, Galois seems to have had a childhood that was both happy and studious.

Galois continued his studies at the Collège Louis-le-Grand in Paris, entering in October, 1823. He found it difficult to adjust to the harsh discipline imposed by the school during the Restoration at the orders of the political authorities and the Church, and, although a brilliant student, he was rebellious. During the early months of 1827, he attended the first-year preparatory mathematics courses taught by H. J. Vernier; this first exposure to mathematics was a revelation for him. He rapidly became bored with the elementary nature of this instruction and with the inadequacies of some of his textbooks and began reading the original works themselves.

After appreciating the difficulty of Adrien-Marie Legendre’s geometry, Galois acquired a solid background from the major works of Joseph-Louis Lagrange. During the next two years, he attended Vernier’s second-year preparatory mathematics courses, then the more advanced ones of L. P. E. Richard, who was the first to recognize Galois’s superiority in mathematics. With this perceptive teacher, Galois excelled in his studies, even though he was already devoting much more of his time to his personal work than to his classwork. In 1828, he began to study some then-recent works on the theory of equations, on number theory, and on the theory of elliptic functions.

This was the time period in which Galois’s first memoir appeared. Published in March, 1829, in the Annales de mathématiques pures et appliquées (annals of pure and applied mathematics), it demonstrated and clarified a result of Lagrange concerning continuous fractions. Although it revealed a certain astuteness, it did not demonstrate exceptional talent.

Life’s Work

In 1828, by his own admission Galois falsely believed—as Niels Henrik Abel had eight years earlier—that he had solved the general fifth-degree equation. Quickly enlightened, he resumed with a new approach the study of the theory of equations, a subject that he pursued until he elucidated the general problem with the aid of group theory. The results he obtained in May, 1829, were sent to the Academy of Sciences by a particularly competent judge, Augustin-Louis Cauchy. Fate was to frustrate these brilliant beginnings, however, and to leave a lasting impression on the personality of the young mathematician.

First, at the beginning of July, his father, a man who had been persecuted for his liberal beliefs, committed suicide. A month later, Galois failed the entrance examination for the Ècole Polytechnique, because he refused to use the expository method suggested by the examiner. Barred from entering the school that attracted him because of its scientific prestige and liberal tradition, he took the entrance examination for the Ècole Normale Supérieure (then called the Ècole Préparatoire), which trained future secondary school teachers. He entered the institution in November, 1829.

At this time he learned of Abel’s death and, at the same time, that Abel’s last published memoir contained several original results that Galois himself had presented as original in his memoir to the Academy. Cauchy, assigned to supervise Galois’s work, advised his student to revise his memoir, taking into account Abel’s research and new results. Galois wrote a new text that he submitted to the Academy in February, 1830, that he hoped would win for him the grand prix in mathematics. However, this memoir was lost upon the death of Joseph Fourier, who had been appointed to study it. Eliminated from the competition, Galois believed himself to be the object of a new persecution by both the representatives of institutional science and society in general. His manuscripts preserve a partial record of the revision of this memoir of February, 1830.

In June, 1830, Galois published in Bulletin des sciences mathématiques (bulletin of mathematical sciences) a short note on the resolution of numerical equations, as well as a much more significant article, “Sur la théorie des nombres” (on number theory). The fact that this same issue contained original works by Cauchy and Siméon-Denis Poisson sufficiently confirms the reputation that Galois had already acquired. The July Revolution of 1830, however, was to initiate a drastic change in his career.

Galois became politicized. Before returning for a second year to the Ècole Normale Supérieure in November, 1830, he had already developed friendships with several republican leaders. Even less able to tolerate his school’s strict discipline than before, he published a violent article against its director in an opposition journal. For this action he was expelled on December 8, 1830.

Left alone, Galois devoted most of his time to political propaganda. He participated in the riots and demonstrations then agitating Paris and was even arrested (but was eventually acquitted). Meanwhile, to a limited degree, he continued his mathematical research. His last two publications were a short note on analysis in the Bulletin des sciences mathématiques of December, 1830, and “Lettre sur l’enseignement des sciences” (letter on the teaching of the sciences), which appeared on January 2, 1831, in the Gazette des écoles. On January 13, he began to teach a public course on advanced algebra in which he planned to present his own discoveries; this project appears not to have been successful.

On January 17, 1831, Galois presented the Academy a new version of his memoir, hastily written at Poisson’s request. However, in Poisson’s report of July 4, 1831, on this, Galois’s most important piece of work, Poisson suggested that a portion of the results could be found in several posthumous writings of Abel and that the rest was incomprehensible. Such a judgment, the profound injustice of which would become apparent in the future, only encouraged Galois’s rebellion.

Arrested again during a republican demonstration on July 14, 1831, and imprisoned, Galois nevertheless continued his mathematical research, revised his memoir on equations, and worked on the applications of his theory and on elliptic functions. After the announcement of a cholera epidemic on March 16, 1832, he was transferred to a nursing home, where he resumed his investigations, wrote several essays on the philosophy of science, and became immersed in a love affair that ended unhappily. Galois sank into a deep depression.

Provoked into a duel under unclear circumstances following this breakup, Galois sensed that he was near death. On May 29, he wrote desperate letters to his republican friends, hastily sorted his papers, and addressed to his friend Auguste Chevalier—but intended for Carl Friedrich Gauss and Carl Gustav Jacob Jacobi—a testamentary letter, a tragic document in which he attempted to outline the principal results that he had attained. On May 30, fatally wounded by an unknown opponent, he was hospitalized; he died the following day, not even twenty-one years of age.

Significance

Èvariste Galois’s work seems not to have been fully appreciated by anyone during his lifetime. Cauchy, who would have been able to understand its significance, left France in September, 1830, having seen only its initial outlines. In addition, the few fragments published during his lifetime did not give an overall view of his achievement and, in particular, did not provide a means of judging the exceptional interest of the results regarding the theory of equations rejected by Poisson. Also, the publication of the famous testamentary letter does not appear to have attracted the attention it deserved.

It was not until September, 1843, that Joseph Liouville, who prepared Galois’s manuscripts for publication, announced officially that the young mathematician had effectively solved the problem, already investigated by Abel, of deciding whether an irreducible first-degree equation is solvable with the use of radicals. Although announced and prepared for the end of 1843, the memoir of 1831 did not appear until the October/November, 1846, issue of the Journal de mathématiques pures et appliquées, when it was published with a fragment on the primitive equations solvable by radicals.

Beginning with Liouville’s edition, which appeared in book form in 1897, Galois’s work became progressively known to mathematicians and subsequently exerted a profound influence on the development of modern mathematics. Also important, although they came too late to contribute to the advancement of mathematics, are the previously unpublished texts that appeared later.

Although he formulated more precisely essential ideas that were already being investigated, Galois also introduced others that, once stated, played an important role in the genesis of modern algebra. Furthermore, he boldly generalized certain classic methods in other fields and succeeded in providing a complete solution and a generalization of problems by systematically drawing upon group theory—one of the most important structural concepts that unified the multiplicity of algebras in the nineteenth century.

Bibliography

Bell, Eric T. Men of Mathematics. New York: Simon & Schuster, 1937. Historical account of the major figures in mathematics from the Greeks to Georg Cantor, written in an interesting if at times exaggerated style. In a relatively brief chapter, “Genius and Stupidity,” Bell describes the life and work of Galois in a tone that both worships and scorns the young mathematician and mixes fact with legend in his discussion.

Boyer, Carl B. A History of Mathematics. New York: John Wiley & Sons, 1968. In this standard and reputable history of mathematics, Boyer devotes a brief section to Galois. Galois is described as the individual who most contributed to the vital discovery of the group concept. The author also assesses Galois’s impact on future generations of mathematicians.

Infeld, Leopold. Whom the Gods Love: The Story of Èvariste Galois. New York: Whittlesey House, 1948. This biography takes great license with the facts (many of which are unknown) of Galois’s life and creates an interesting, if fictional, account. The author, maintaining that biography always mixes truth and fiction, puts Galois’s life in the historical context of nineteenth century France by creating scenes and dialogues that might have occurred.

Kline, Morris. Mathematical Thought from Ancient Times to Modern Times. New York: Oxford University Press, 1972. In this voluminous work, the author surveys the major mathematical creators and developments through the first few decades of the twentieth century. The emphasis is on the leading mathematical themes rather than on the men. The brief section on Galois gives some biographical information and discusses the mathematician’s work in finite fields, group theory, and the theory of equations.

Struik, Dirk J. A Concise History of Mathematics. Vol. 2 in The Seventeenth Century-Nineteenth Century. New York: Dover, 1948. In this book devoted to a concise overview of the major figures and trends in mathematics during the time period covered, a brief section is devoted to Galois. The author spends approximately equal time discussing Galois’s life and major achievements, and views the mathematician both as a product of his times and as a unique genius.

Tota Rigatelli, Laura. Evariste Galois, 1811-1832. Translated from the Italian by John Denton. Boston: Birkhäuser Verlag, 1996. Brief biography based on new research offering a more accurate account of Galois’s life than previous biographies. Includes a chapter describing Galois’s mathematical work and a comprehensive bibliography.