Is $K(E_{tors})/K$ always infinite degree extension

Question

This question is related to complex multiplication and Kronecker's young dream. 　

Let $K$ be a imaginary quadratic number field and let $E/\Bbb{C}$ be an elliptic curve which has complex multiplication over $R_K$(ring of integers of $K$).

Let $E_{tors}$ be all coordinates of all of the torsion points of $E$. Then, is $K(E_{tors})/K$ always infinite degree extension ?

　My thoughts : This is true, to prove this, suppose $K(E_{tors})/K$ be degree $n$ extension. I want to find an element of its minimal polynomial which has degree more than $n＋1$.But I don't have confident I can find such an element.

Answer 1

Yes it does. There’s a nice argument based on the perfectness of the Weil pairing that you can find in Silverman. In summary, if a field contains all the $n$ torsion points, then we can apply the Weil pairing to them to produce a primitive $n$-th root of unity in that field. Since this works for any $n$, the field must contain every cyclotomic field and thus must be of infinite degree over $K$.

Answer 2

Yes, it is certainly always an infinite extension. It is difficult to give a proof for this question without knowing what theorems you know and don't know, but for example the prime-to-$p$-torsion subgroup of an elliptic curve over a number field injects into the torsion over $\mathbb{F}_p$ for sufficiently large $p$, which rules out the torsion subgroup being infinite over any number field.