  Given any positive integer n, we prove the existence of infinitely many right
triangles with area n and side lengths in certain number fields. This
generalizes the famous congruent number problem. The proof allows the explicit
construction of these triangles; for this purpose we find for any positive
integer n an explicit cubic number field Q(\lambda) (depending on n) and an
explicit point P_\lambda of infinite order in the Mordell-Weil group of the
elliptic curve Y^2=X^3-n^2*X over Q(\lambda).
