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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