Let $E/\mathbb{C}$ be an elliptic curve. In exercise 3.8a) of chapter III, Silverman states that there is a lattice $L \subset \mathbb{C}$ and a complex isomorphism of groups $\mathbb{C}/L \cong E(\mathbb{C})$. From this, he asks the reader to prove that one has $deg([m])=m^2$ and $E[m] \cong \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/m\mathbb{Z} $, where $[m]$ denotes the usual multiplication by $m$ map in the elliptic curve.
Now, in exercise 3.8b), Silverman writes:
Let $E/K$ be an elliptic curve with $char(K) = 0$. Using (a), prove that $deg[m] = m^2$. [Hint: If $K$ can be embedded in $\mathbb{C}$, there is no problem. Reduce to this case.]
Hence, my question is: How can one reduce the problem to the case where $K \subset \mathbb{C}$?
I am (vaguely) aware about the existence of techniques like the Lefschetz principle for algebraic geometry type problems, which would solve this issue, but surely this is overkill for such a problem. I also know that we can't embed any field of characteristic $0$ into $\mathbb{C}$ (something like $\mathbb{Q}(X)$ where $X$ is a set of cardinality $> 2^{\aleph_0}$ should be an example), so I don't really see how to attack this problem. Any help?