2.6.TOPOLOGICAL INTERPRETATION 9 implies E has a K-rational place, thus is a rational field K(x), e.g. by [59, 1.6.3]. Write t = f(x) with / G K(X), then / is the desired function. DEFINITION 2.9. Let C be a conjugacy class of G, n the order of G, and £n a primitive root of unity. For 7 an automorphism of Q let m be with 7(Cn) Cm- Then the class C is said to be K-rational if Cm = C for each 7 G Gal(^|K). This property is well known to be equivalent to x(C) G if for all irreducible characters X of G. In particular, if the character values of the classes in Ti as above are in K, and B C K, then (S, W) is if-rational. 2.6. Topological interpretation Though we do not really need it here, it might help understanding some argu- ments in this paper if one also has the geometric interpretation of a branch cycle description in mind. Suppose that K is a subfield of the complex numbers C. As C(t) H L = K(t) (see [5, Corollary 2, V, §4]), we get Gal(CL|C(t)) ^ Gal(L\K(t)) by restriction. For any holomorphic covering a : A » B of Riemann surfaces, denote by a* : Ai(B) c -^ .M(A) the natural inclusion of the fields of meromorphic functions on B and A. Associated to CE\C(t) is a branched, holomorphic covering 7r : S PX(C) of degree n = [-E : i^(t)] with * S a connected Riemann surface and P1(C) the Riemann sphere, such that the extension A/!(5f)|7r*(A^(P1(C))) can be identified with CE\C{t). Observe that ^ ( P ^ C ) ) ^ C(t). Let B = {pi,p2, ,Pr} be the set of branch points of ir. Fix p0 £ P1(C) \ B, and denote by Q the fundamental group 7Ti(P1(C) \S,po). Then Q acts transitively on the points of the fiber 7r-1(p0) by lifting of paths. Fix a numbering 1, 2,... , n of this fiber. Thus we get a homomorphism Q Sn of Q into the symmetric group Sn. By standard arguments, the image of Q can be identified with the geometric monodromy group G defined above, thus we write G for this group too. This identification has a combinatorial consequence. Choose a standard homo- topy basis of P1(C) \ B as follows. Let 7^ be represented by paths which wind once around p^ clockwise, and around no other branch point, such that 7172 -7 r = 1. Then 71,72, •, 7r-i freely generate Q. Po
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