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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