The Mordell-Weil theorem for elliptic curves

Abstract

The main goal of this thesis is to prove the Mordell-Weil theorem for elliptic curves. That is, that the set of $K$-rational points on an elliptic curve form a finitely generated abelian group under the natural addition, where $K$ is a number field. To do this we first define what an elliptic curve is and prove some elementary results. This includes a short introduction into Galois theory for arbitrary algebraic field extensions. We then give a sufficient condition for an abelian group to be finitely generated. Under the assumption that $E(\mathbb{Q})/2E(\mathbb{Q})$ is finite, we prove the theorem for the special case $K = \mathbb{Q}$. The rest of the thesis will consist of proving that $ E(K)/mE(K)$ is finite, where $m$ is a positive integer. This involves tools and concepts from field theory and algebraic number theory.