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
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