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

Solvability of Equations 227
where is an algebraic function.
A priori the solution of equation (A.11) could be expressed by very
complicated formulae. Theorem 3 says that this is nowhere the case.
Either the equation has sufficiently simple roots, or in general it cannot
be solved by generalized quadratures.
Liouville found a series of results of this type. The common idea is the
following: simple equations have either simple solutions, or in general have
no solutions in a given class (by quadratures, by elementary functions,
etc.).
The strategy of the proof in Liouville’s theory is the following: prove
that if a simple equation has a solution which is represented by a complicated
formula then this formula can be always simplified.
Liouville, undoubtedly, was inspired by the results by Lagrange, Abel,
and Galois on the non-solvability by radicals of algebraic equations. Differently
from the Galois theory, Liuoville’s theory does not involve the
notion of the group of automorphisms. Liouville, however, uses, in order
to simplify his formulae, ‘infinitely small automorphisms’.
Let us return to Theorem 2 on the integrability of algebraic functions.
The following corollary follows from this theorem.
COROLLARY. If the integral of an algebraic function A is a generalized
elementary function then the differential form has some unavoidable
singularities on the Riemann surface of the algebraic function A.
It is well known that on every algebraic curve with positive genus
there exist non-singular differential forms (the so called abelian differentials
of first type). It follows that algebraic functions whose Riemann
surfaces have positive genus are not, in general, integrable by generalized
elementary functions.
This was already known by Abel, who discovered it as he was proving
the non-solvability by radicals of a fifth-degree generic equation. Observe
also that the Abel proof of the non-solvability by radicals is based on topological
arguments. I do not know whether the topological properties of the
Riemann surfaces of functions representable by generalized quadratures
are different from those of the Riemann surfaces of generalized elementary
functions. Indeed, I am unable to prove through topological arguments
that the integral of an algebraic function is not an elementary function:
each one of such integrals is by definition a function representable by
generalized quadratures. However, if an algebraic function depends on
a parameter its integral may depend on the parameter in an arbitrarily
complicated manner. One can prove that the integral of an algebraic