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

a2

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,

ao 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.