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Summary Article: Galois, Evariste (1811-1832) from The Hutchinson Dictionary of Scientific Biography

Place: France

Subject: biography, maths and statistics

French mathematician who, building on the work of Joseph Lagrange, Karl Gauss, Niels Abel, and Augustin Cauchy, greatly extended the understanding of the conditions in which an algebraic equation is solvable and, by his method of doing so, laid the foundations of modern group theory.

Galois was born in the village of Bourg-la-Reine, on the outskirts of Paris, on 25 October 1811. His family was of the highly respectable bourgeois class that came into its own with the restoration of the monarchy after the Napoleonic empire. His father was the headmaster of the local boarding school and mayor of the town; his mother came from a family of jurists, and it was she who gave Galois his early (chiefly classical) education. He first attended school at the age of 12, when he was sent to the Collège Louis-le-Grand in Paris as a fourth-form boarder in 1823. Another four years passed before Galois, who gained notoriety for resisting the strict discipline imposed at the school, had his mathematical imagination fired by the lectures of H J Vernier. He soon became familiar with the works of Legrange and Adrien Legendre and by 1828 he was busy mastering the most recent work on the theory of equations, number theory, and elliptic functions. He quickly came to believe that he had solved the general fifth-degree equation. Like Abel before him, he discovered his error, and that discovery launched him on his search - ultimately successful and ultimately of momentous consequence for the future of mathematics - for a solution to the problem of the solubility of algebraic equations generally. By 1829 he had progressed far enough to interest Augustin Cauchy, who presented Galois' early results to the Academy of Science.

In a remarkably short period of time Galois, at the age of 17, had arrived very near the apex of existing mathematical thought. Then the first of the emotional disruptions that were to darken the few remaining years of his life occurred. His father, the victim of a political plot that unjustly discredited him, committed suicide in July 1829. A month later Galois, partly from his fiery impatience with the examiners' instructions, failed to gain entrance to the Ecole Polytechnique. He had therefore to be content with entering, in the autumn of 1829, the Ecole Normale Supérieure.

It was then that Galois learned that the ideas that were contained in the paper presented to the Academy of Science by Cauchy were not original. Shortly before, very similar ideas had been published by Abel in his last paper. Encouraged by Cauchy, Galois began to revise his paper in the light of Abel's findings. He presented the revised version to the academy in February 1830, with high hopes that it would gain him the Grand Prix. To his dismay the academy not only rejected his paper, but the examiner lost the manuscript. Galois was indignant at this ill-treatment, but four months later he succeeded in having a paper on number theory published in the prestigious Bulletin des sciences mathématiques. It was this paper that contained the highly original and diverting theory of ‘Galois imaginaires’.

Thus, when the July Revolution that drove Charles X off the throne shocked French society, Galois had reason to be proud of his barely acknowledged mathematical genius and cause to feel estranged, both on his own and his father's account, from the stuffy officialdom that had cast its pall over France from 1815. He joined the revolutionary movement. In the next year he was twice arrested, in May 1831 for proposing a regicide toast at a republican banquet (Louis-Philippe having taken Charles's place as king) and again in July for taking part in a republican demonstration. For the second offence Galois was imprisoned for nine months. In prison he continued his mathematical research. But shortly after his release he became involved in a duel, perhaps from political reasons, perhaps from complications arising from a love affair. The event remains mysterious. Galois, it is evident, expected to be killed. On 29 May 1832, in a letter written in feverish haste, he outlined the principal results of his mathematical inquiries. He sent the letter to his friend Auguste Chevalier, almost certainly in the expectation that it would find its way to Karl Gauss and Karl Jacobi in Germany. The duel took place on the following morning. Galois was severely wounded in the stomach and died in hospital on 31 May 1832.

So brief was Galois's life - he was only 20 when he died - and so dismissively were his ideas treated by the academy, that he was known to his contemporaries principally as a rather headstrong republican agitator. Cauchy, who alone sensed the importance of what Galois was doing, was out of France after 1830 and did not see any of the revisions or developments of Galois's first paper. Moreover, in his letter to Chevalier, Galois had time only to put down in concise form - without demonstrations - his most important conclusions. When he died, therefore, Galois's work amounted to fewer than a hundred pages, much of it fragmentary and nearly all of it unpublished.

The honour of rescuing Galois from obscurity belongs, in the first instance, to Joseph Liouville, who in 1843 began to prepare his papers for publication and informed the academy that Galois had provided a convincing answer to the question whether first-degree equations were solvable by radicals. Finally, in 1846, both Galois's 1831 paper and a short notice on the solution of primitive equations by radicals were (thanks to Liouville) published in the Bulletin.

Galois's achievement, put tersely, was to arrive at a definitive solution to the problem of the solvability of algebraic equations, and in doing so to produce such a breakthrough in the understanding of fields of algebraic numbers and also of groups that he is deservedly considered as the chief founder of modern group theory. What has come to be known as the Galois theorem made immediately demonstrable the insolubility of higher-than-fourth-degree equations by radicals. The theorem also showed that if the highest power of x is a prime, and if all other values of x can be found by taking only two values of x and combining them using only addition, subtraction, multiplication, and division, then the equation can be solved by using formulae similar in principle to the formula used in solving quadratic equations.

Galois's work involved groups formed from the arrangements of the roots of equations and their subgroups, groups that he fitted into each other rather on the analogy of the Chinese box arrangement. This, his most far-reaching achievement for the subsequent development of group theory, is known as Galois theory. Along with other such terms - Galois groups, Galois fields, and the Galois theorem - it bears testimony to the lasting influence of his rejected genius.

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