Preface to the First German Edition

xm

has the solution

xx = ^ - l + v^-h \/3 + 2/2 + ^ 3 - 2 ^ / 2 + y/-l - V2.

Of much greater significance than such solutions is the method

that Galois discovered, which was unorthodox, indeed revolutionary,

at the time, but today is quite usual in mathematics. What Galois

did was to establish a relationship between two completely different

types of mathematical objects and their properties. In this way he

was able to read off the properties of one of these objects, namely

the solvability of a given equation and the steps in its solution, from

those of the corresponding object.

But it was not only the principle of this approach that benefited

future mathematics. In addition, the class of mathematical objects

that Galois created for the indirect investigation of polynomial equa-

tions became an important mathematical object in its own right, one

with many important applications. This class, together with similar

objects, today forms the foundation of modern algebra, and other

subdisciplines of mathematics have also progressed along analogous

paths.

The object created by Galois that corresponds to a given equa-

tion, called today the Galois group, can be defined on the basis of

relations between the solutions of the equation in the form of iden-

tities such as x\ = X2 + 2. Concretely, the Galois group consists of

renumberings of the solutions. Such a renumbering belongs to the

Galois group precisely if every relationship is transformed by this

renumbering into an already existing relationship. Thus for the case

of the relation x\ = X2 + 2 in our example, the renumbering corre-

sponding to exchanging the two solutions x\ and X2 belongs to the

Galois group only if the identity x\ — x\ + 2 is satisfied. Finally,

every renumbering belonging to the Galois group corresponds to a

symmetry among the solutions of the equation. Moreover, the Galois

group can be determined without knowledge of the solutions.

The Galois group can be described by a finite table that is ele-

mentary but not particularly elegant. Such a table is called a group

table, and it can be looked upon as a sort of multiplication table, in