Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite? What else can we say about it? Does it look more like $\bigoplus \mathbb Z$ or $\mathbb C/\Gamma$? Is it divisible? (I don't think so). I apologize that the questions are not precise, I just need to know which results do we have in this direction.