Mordell-Weil theorem

The Mordell-Weil theorem states that for an elliptic curve defined over a number field (a type of mathematical system), the group of rational points on the curve forms a finitely generated abelian group. This means that all rational solutions can be built up from a finite set of basic points and their repeated addition, much like how whole numbers can be generated from prime factors. Essentially, this theorem assures that the solutions are not infinite and can be systematically classified, which is fundamental for understanding the structure of these solutions in number theory and cryptography.