Mordell-Weil theorem

Statement
Any elliptic curve group over a number field (viz a finite extension of $$\mathbb{Q}$$) is finitely generated.

Actually the result is a bit more general: it says that for any Abelian variety over a number field, the divisor class group is finitely generated. Since we already know it is an Abelian group, this basically, by the structure theorem for Abelian groups, tells us that the divisor class group is the product of a free Abelian group and a finite group (the finite group is called the torsion part).

Related results

 * Mazur's theorem
 * Lutz-Nagell theorem