This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.