Preface to the First German Edition xv Thus starting with a given equation xn + cin-ix71'1 H h a\x + a0 = 0, one forms the smallest set of numbers that contains all quantities, such as a 2 2 . ax + a0, a0 that can be obtained from the coefficients of the equation using suc- cessive basic arithmetic operations. Then one obtains an enlarged set of numbers that is of particular use in studying the given equation by allowing in one's calculations, in addition to the coefficients of the equation, the solutions #i, X2,.... This set is therefore formed of all numbers that can be obtained from expressions of the form, for example, a o 2 , —xx - a2x2 + ai. a2 If it is now possible to represent the solutions of the given equation by nested expressions involving radicals, then one can obtain additional fields of numbers by allowing in addition to the coefficients some of these nested radicals. Thus every solution of an equation corresponds to a series of nested fields of numbers, and these can be found, accord- ing to the main theorem of Galois theory, by analysis of the Galois group. Thus by an analysis of the Galois group alone, one can answer the question whether the solutions of an equation can be expressed in radicals. £ J~-*~ *—~, y~~J ~ y~*~ ~ ~*~* ^/*T*/* / - # ~ fr» cites c,U-» *« f£~»J~- *~ r- ^^.AS,^ ** rr Figure 0.2. Evariste Galois and a fragment from his last let- ter. In this passage he describes how a group G can be de- composed with the help of the subgroup H. See Section 10.4.

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