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JORDAN, CAMILLE.

Commentaire sur Galois.

lyn52165

Leipzig, B.G. Teubner, 1869. 8vo. Bound in later half cloth with gilt lettering to spine. In "Mathematische Annalen", Volume 1., 1869. Entire volume offered. Library stamp to front free end-paper and title-page. Internally a few light brown spots.. Pp. 141-160. [Entire volume: Pp. IV-636.] ¶ First printing of the very first issue of the influential mathematical journal "Mathematische Annalen" containing Camille Jordan famous paper on Galois group theory.

"When Jordan started his mathematical career, Galois's profound ideas and results (Which had remained unknown to most mathematicians until 1845) were still very poorly understood. Jordan was the first to embark on a systematic development of the theory of finite groups and of its applications in the directions opened by Galois. Chief among his first results were the concept of composition series and the first part of the famous Jordan-Hölder theorem, proving the invariance of the system of indexes of consecutive groups in any composition series (up to their ordering). He also was the first to investigate the structure of the general linear group and of the "classical" groups over a prime finite field, and he very ingeniously applied his results to a great range of problems; in particular, he was able to determine the structure of the Galois group of equations having as roots the parameters of some well-known geometric configurations (the twenty-seven lines on a cubic surface, the twentyhyphen;eight double tangents to a quartic, the sixteen double points of a Kummer surface, and so on).
Another problem for which Jordan's knowledge of these classical groups was the key to the solution, and to which he devoted a considerable amount of effort from the beginning of his career, was the general study of solvable finite groups. From all we know today (in particular about p groups. a field which was started, in the generation following Jordan, with the Sylow theorems) it seems hopeless to expect a complete classification of all solvable groups which would characterize each of them, for instance by a system of numerical invariants. Perhaps Jordan realized this; at any rate he contented himself with setting up the machinery that would automatically yield all solvable groups of a given order n." (DSB)

"There have been few mathematicians with personalities as engaging as that of Galois, who died at the age of twenty years and seven months from wounds received in a mysterious duel. He left a body of work-for the most part published posthumously less than 100 pages, the astonishing richness of which was revealed in the second half of the nineteenth century. Far from being a cloistered scholar, this extraordinarily precocious and exceptionally profound genius had an extremely tormented life. A militant republican, driven to revolt by the adversity that overwhelmed him and by the incomprehension and disdain with which the scientific world received his works, to most of his contemporaries he was only a political agitator. Yet in fact, continuing the work of Abel, he produced with the aid of group theory a definitive answer to the problem of the solvability of algebraic equations, a problem that had absorbed the attention of mathematicians since the eighteenth century; he thereby laid one of the foundations of modern algebra. The few sketches remaining of other works that he devoted to the theory of elliptic functions and that of Abelian integrals and his reflections. on the philosophy and methodology of mathematics display an uncanny foreknowledge of modern mathematics.

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