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