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

Appendix (V.I. Arnold)
The topological arguments for the different types of non-solvability (of
equations by radicals, of integrals by elementary functions, of differential
equations by quadratures etc.) can be expressed in terms of very precise
questions.
Consider, for example, the problem of the integration of algebraic
functions (i.e, the search for Abelian integrals). The question in this
example consists in knowing whether these integrals and their inverse
functions (for example, the elliptic sinus) are topologically equivalent to
elementary functions.
The topological equivalence of two mappings and of M to N means
the existence of a homeomorphisms of M into M and a homeomorphism
of N into N which transform into i.e, such that
The absence amongst the objects of a class B of an object topologically
equivalent to the objects of a class A means the topological nonreducibility
of A to B (of the Abelian integrals and of the elliptic functions
to the elementary functions, etc.).
In my lectures in the years 1963–1964 I expounded the topological
proof of all three aforementioned versions of the Abel problems (cf., [6],
[7]), but the book extracted from my lectures contains only the topological
proof of the non-solvability by radicals of the algebraic equations of degree
5.
Since I am unable to give references of the unpublished proofs of the
two remaining enunciations of the topological non-solvability, here it is
convenient to call them ‘problems’. I underline only that, although the
non-solvability of every problem follows from the non-solvability in the
topological sense explained above, the assertion about the topological
non-solvability is stronger and it is not proved by means of calculations,
showing the non-existence of the formulae sought.
This topological point of view of the non-solvability is also applied
to many other problems; for example, to the results by Newton [4], [5],
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