As it was pointed out by Lang [4, p. 245] and others, certain
finiteness results for diophantine equations over algebraic number
fields can be extended, by using deep algebraic geometrical arguments,
to rather general cases when the ground domain of unknowns is a
finitely generated field or a finitely generated subring of it.
The purpose of this paper is to establish a surprisingly elementary
method, through a concrete equation, to obtain these kind of general
Let f(X, Y) and g(X, Y) he binary forms (homogeneous polynomials in two
variables) with complex coefficients of degree m and n, respectively.
The binary form fg splits into linear factors (over C) and in the
sequel, we suppose that the linear factors are pairwise
non-proportional. Let K be a finitely generated subfield of C. Then K
can be written in the form Q(z(1),.,z(q), u), where z(1),...,z(4) is a
transcendence basis of K and we may assume without loss of generality
that the element u is integral over the polynomial ring