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Let K be a number field, ax the ring of integers of K. Suppose we are given a projective curve 2, and a morphism 4 : Z 4 Pf(C) (where Pf(C) is the projective line) such that 4 and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z) "/K(Pt(C))) (where K(Z) A is the normal closure of K(Z) over K @ ' ( C ) ) ) . In Section 1 we investigate (for our needs) the combinatorial and group theoretical aspects of the situation where

The field of definition of function fields and a problem in the reducibility of polynomials in two variables