Abel's theorem in problems and solutions - School of Mathematics

222 Appendix by Khovanskii
of this Appendix and F. Aicardi for the translation into English.
A.1 Explicit solvability of equations
Some differential equations possess ‘explicit solutions’. If it is the case
the solution gives itself the answer to the problem of solvability. But in
general all attempts to find explicit solutions of equations turn out to be
in vain. One thus tries to prove that for some class of equations explicit
solutions do not exist. We must now define correctly what this means
(otherwise, it will not be clear what we really wish to prove). We choose
the following way: we distinguish some classes of functions, and we say
that an equation is explicitly solvable if its solution belongs to one of
these classes. To different classes of functions there correspond different
notions of solvability.
To define a class of functions we give a list of basic functions and a
list of allowed operations.
The class of functions is thus defined as the set of all functions which
are obtained from the basic functions by means of the allowed operations.
EXAMPLE 1. The class of the functions representable by radicals.
The list of basic functions: constants and the identity function (whose
value is equal to that of the independent variable).
The list of the allowed operations: the arithmetic operations (addition,
subtraction, multiplication, division) and the root extractions
of a given function
The function is an example of a function
representable by radicals.
The famous problem of the solvability of the algebraic equations by
radicals is related to this class. Consider the algebraic equation
in which are rational functions of one variable. The complete answer
to the problem of the solvability of the equation (A.10) by radicals consists
in the Galois theory (see §A.8).
Note that already in the simplest class, that in Example 1, we encounter
some difficulties: the functions we deal with are multi-valued.
Let us see exactly, for example, what is the sum of two multi-valued
analytic functions and Consider an arbitrary point one of the
germs of the function at the point and one of the germs of the