In two papers [6; 7], Hurwitz dealt with unbounded coverings of compact Riemann surfaces Y. He was able to determine the number of ("geometrically") different coverings in case all resp. all but one ramified points of Y split in a single point of order two and points of order one. In the present paper we take up this question and ask for upper and lower bounds of the number of different unbounded coverings of Y which have a prescribed ramification type. In case the degree of the covering (=

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... er of sheets) equals n we find a lower bound that is very roughly (n\)2e+r~3 and an upper bound that is very roughly (n!)2e+r_1 where gis the genus of the compact Riemann surface Y and r is the number of ramified points of Y. The tools used in getting these estimates are some results on permutation groups like: (i) Every element of the alternating group A" of n elements is a commutator. (ii) Let n 7^ 4 and let a be an element of the symmetric group S" of n elements that is not the identity. IfaeA" ($ An), then every element of A" (S") can be written as a product of at most n conjugates of a. (iii) Given an element of S" that is not the identity. Then there is another element of S" which, together with the first one, generates Sn. The second part of the paper deals with extensions of algebraic function fields of one variable whose local splitting is prescribed. It is clear that nonisomorphic field extensions correspond to different unbounded coverings, but different unbounded coverings may belong to isomorphic field extensions. Therefore, an upper bound for the number of different unbounded coverings (with prescribed ramification type) is also an upper bound for the number of nonisomorphic field extensions (with prescribed local splitting). Surprisingly, it turns out that the above mentioned lower bound (concerning coverings) also serves in general as a lower bound for the number of nonisomorphic field extensions. This result is of some interest in connection with a potential class field theory of arbitrary ramified field extensions. In the third part, we determine completely the algebraic