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

224 Appendix by Khovanskii
if together with every set of functions it also contains a function
satisfying the equation
EXAMPLE 4. If in the definition of the class of elementary functions
we add the operation of solution of algebraic equations, we obtain the
class of the generalized elementary functions.
EXAMPLE 5. The class of functions representable by generalized
quadratures contains the functions obtained from the class of functions
representable by quadratures by adding the operation of solution of algebraic
equations.
A.2 Liouville’s theory
The first exact proofs of the non-solvability of some equations neither by
quadratures nor by elementary functions were obtained by Liouville in
the middle of the XIX century. Here we briefly expound his results.
The reader can find a wider exposition of the Liouville method and
of the works on analogous subjects by Chebychev, Mordukai-Boltovski,
Ostrovski, and Ritt in book [1].
First of all Liouville showed that the classes of functions in Examples
2–5 can be constructed in a very simple way. Indeed, the set of basic
elementary functions seems to be very large. Moreover, in the definition
of this class one encounters some algebraic difficulties owed to the composition
operation. Liouville at first proved that one can reduce a great
deal the lists of basic functions, in one half of the cases leaving in it only
the constants, and in the remaining cases leaving only the constants and
the identity function. Secondly, he proved that in the list of the allowed
operations the composition is superfluous. One can define all the necessary
operations using only arithmetic operations and differentiation. This
fact plays an essential role for the algebraization of the problem of the
differential fields numerability.
Let us formulate the corresponding definitions in differential algebra.
A field of functions F is called a differential field if it is closed with
respect to the differentiation, i.e., if then One can also
consider the abstract differential fields, i.e., the field in which one can
define a supplementary differentiation operation, satisfying the Leibniz
identity