X l l Prefaces to the German Editions can be shown to have the solution 4 + 2\/2+ V4-2/ 2 + V - V^4 + 2 v ^ - \/4 - V2 + 2v/2 ^ 3 + 2 ^ + 2 ^ 3 - 2^3 - 2 With the almost simultaneous discovery of formulas for solving third- and fourth-degree equations came the inevitable problem of finding similar formulas for equations of higher degree. To accomplish this, the techniques that were used for the cubic and quartic equations were systematized, already in Cardano's time, so that they could be applied to equations of the fifth degree. But after three hundred years of failure, mathematicians began to suspect that perhaps there were no such formulas after all. This question was resolved in 1826 by Niels Henrik Abel (1802- 1829), who showed that there cannot exist general solution formulas for equations of the fifth and higher degree that involve only the usual arithmetic operations and extraction of roots. One says that such equations cannot be solved in radicals. The heart of Abel's proof is that for the intermediate values that would appear in a hypothetically existing formula, one could prove corresponding symmetries among the various solutions of the equation that would lead to a contradic- tion. Galois Theory. A generalization of Abel's approach, which was ap- plicable to all polynomial equations, was found a few years later by the twenty-year-old Evariste Galois (1811-1832). He wrote down the results of his researches of the previous few months on the evening before he was killed in a duel. In these writings are criteria that allow one to investigate any particular equation and determine whether it can be solved in radicals. For example, the solutions to the equation x5 - x - 1 = 0 cannot be so expressed, while the equation x5 + 15x - 44 = 0

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