Évariste Galois

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Galois, Évariste

 

Born Oct. 26, 1811, in Bourg-la-Reine, near Paris; died May 30, 1832, in Paris. French mathematician whose research had an exceptionally strong influence on the development of algebra.

Galois studied in the Lycée Louis le Grand, and by the time he graduated he was already doing original work in mathematics. In 1830 he entered the Ecole Normale and was expelled from it in 1831 for political reasons. The beginning of Galois’s political activities dates to this period: he belonged to the secret republican society Friends of the People. He was imprisoned twice for public action against the royal regime. Almost immediately after his release, at the age of 20, he was killed in a duel, which from all appearances was provoked by his political enemies.

Galois’s mathematical legacy comprises a small number of very concisely written works that were not understood by his contemporaries. Galois, in effect, constructed the entire theory of finite fields (today called Galois fields). In a letter to a friend, written on the eve of the duel, Galois formulated the basic theorems on integrals of algebraic functions, which were rediscovered much later in the works of B. Riemann. Galois’s principal contribution was the formulation of a complex of ideas, which he arrived at as a result of continuing the research on the solvability of algebraic equations by radicals begun by J. Lagrange, N. Abel, and other mathematicians. The Galois theory constructed as a result of this, by establishing the description of extensions of fields in terms of groups resembling a description of the symmetries of a polyhedron, reduces questions concerning fields to questions of the theory of groups (which originated with this development).

WORKS

Sochineniia. Moscow-Leningrad, 1936. (Translated from French.)

REFERENCES

Infeld, L. Evarist Galua: Izbrannik bogov. [Moscow] 1958. (Translated from English.)
Dalmas, A. Evarist Galua, revoliutsioner i matematik. Moscow, 1960. (Translated from French.)

A. I. SKOPIN

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The point P is called a Galois point for C if the field extension C(C)/[K.sub.P] is Galois.
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In this section we mention some results concerning the antitone Galois connections and their relationship with the closure systems in complete lattices.
If [A.sup.G] [subset or equal to] A is a Galois extension, then the following hold:
And in the chapters on Bichat (chapter 4), Davy (chapter 5), and Galois (chapter 6) that follow, Chai traces out analogous distinctions between degrees of reflexivity, ranging from Bichat's attempt to develop a new theory of vitality, to Galois' more ambitious field theory; Galois theory could be extended to include new members in a group that are not yet known--a group defined by a "principle of containment" rather than an account of its elements (147).
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Galois's most important ideas had in fact been submitted in a paper to the French Academy three years earlier.
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We also show that when k = Q, the field of rational numbers, there exist crossed product k(t) and k((t))-division algebras of index [p.sup.3] (for any prime p) whose Galois maximal subfields all have the nonabelian dihedral-type group.