  Let X be an algebraic curve over Q and t a non-constant Q-rational function
on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a
point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the
number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this
conjecture in the special case when t defines a geometrically abelian covering
of the projective line, and the critical values of t are all rational. This
implies, in particular, that our conjecture follows from a famous conjecture of
Schinzel.
