Mordell-Weil Theorem

For Elliptic Curves over the Rationals, $\Bbb{Q}$, the number of generators of the set of Rational Points is always finite. This theorem was proved by Mordell in 1921 and extended by Weil in 1928 to Abelian Varieties over Number Fields.

References

Ireland, K. and Rosen, M. ``The Mordell-Weil Theorem.'' Ch. 19 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 319-338, 1990.