XIV Prefaces to the German Editions which each entry is the result of operating on two elements of the Ga- lois group in succession. An example is shown in Figure 0.1. What is significant about the Galois group, and its corresponding group table, is that it always contains the information about whether, and if so, how, the underlying equation can be solved in radicals. To be sure, the proof of this in a concrete application can be quite involved nev- ertheless, it can always be accomplished in a finite number of steps according to a fixed algorithm. A B C D E F G H I J A B C D E F G H I J A B C D E F G H I J B C D E A G H I J F C D E A B H I J F G D E A B C I J F G H E A B C D J F G H I F J I H G A E D C B G F J I H B A E D C H G F J I C B A E D I H G F J D C B A E J I H G F E D C B A Figure 0.1. The Galois group of the equation x5 — 5x + 12 is represented as a table by means of which the solvability in rad- icals can be determined by purely combinatorial means. This equation will be considered in detail in Section 9.17. Equa- tions of the fifth degree that are not solvable in radicals have tables of size 60 x 60 or 120 x 120. Today, Galois's ideas are described in textbooks in a very ab- stract setting. Using the class of algebraic objects that we previously mentioned, it became possible at the beginning of the twentieth cen- tury to reformulate what has come to be called Galois theory, and indeed in such a way that the problem itself can be posed in terms of such objects. More precisely, the properties of equations and their solution can be characterized in terms of associated sets of numbers whose common characteristic is that they are closed under the four basic arithmetic operations. These sets of numbers are called fields.

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