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Galois Theory

  Galois theory, in mathematics, originated in response to the problem of solving polynomials. Methods of solving quadratics (polynomials of the form ax2 + bx + x = 0) were known, at least in part, to the ancient Babylonians, and are now commonly taught in school. A method of solving cubics (equations of the form ax3 + bx2 + cx + d = 0) was discovered (probably rediscovered) by Tartaglia (Niccolo Fontana; (1500? - 1557)) and published by , Girolamo Cardano (1501 - 1576) in 1545 in his Ars Magna, which also included a method to solve quartics (equations of the form ax4 + bx3 + cx2 + dx + e = 0) discovered by , Ludovico Ferrari (1522 - 1565). The obvious next step was to find a solution of the same kind for the quintic; many mathematicians attempted to do so over the next 280 years, with a growing feeling that the task was impossible. This was finally proved to be the case by , Niels Henrik Abel (1802 - 1829) in 1824.

So the question mathematicians were now considering was to determine when a given equation could be solved in the same way as the quadratic, cubic and quartic. , Évariste Galois (1811 - 1832) was a French mathematician who, during a life which sounds like the plot of a romantic novel, was ignored by the Academy of Sciences in Paris, who failed to understand his highly original and, for the time, highly abstract work. After his death (in a duel over a woman) an extremely elegant solution to this problem was found among his papers. The method is to use the theory of groups in a very advanced way to attack the problem: and the concept of a group was one which Galois had to invent for the purposes. His work was really the beginning of abstract algebra. Many other problems in mathematics can be reduced to the question of solving a polynomial equation, so Galois theory has thrown light in many areas of the subject. SMcL



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