Abstract:
In the topological Galois theory we consider functions representable by quadratures as multivalued analytical functions of one complex variable. It turns out that there are some topological restrictions on the way the Riemann surface of a function representable by quadratures can be positioned over the complex plan. If a function does not satisfy these restrictions, then it cannot be represented by quadratures.
There are the following topological obstructions to the representability of functions
by quadratures. Firstly, functions representable by quadratures, can have no more than countable set of singular points in the complex plane. (However, even for the simplest functions representable by quadratures the set of singular points can be everywhere dense.) Secondly, the monodromy group of a function representable by quadratures is necessarily solvable. (However, even for the simplest functions representable by quadratures the monodromy group can have the cardinality of the continuum.)
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