Torsion points of an elliptic curve over infinite extension How can I see that, if $E$ is an elliptic curve over $\mathbb{Q}$, than $$E(\overline{\mathbb{Q}})_{tors}=E(\mathbb{C})_{tors}=(\mathbb{Q}/\mathbb{Z})^2,$$ and that $E(\overline{\mathbb{Q}})/E(\overline{\mathbb{Q}})_{tors}$ has infinite rank? 

The infinitude of the rank is somewhat subtle. You can see a proof in this article of Frey and Jarden (see Section 2).
