Let $E^{\prime}$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a prime and suppose we have a map $\rho_{E^\prime}: G_{\mathbb{Q}}\rightarrow PGL_2(\mathbb{Z}_p)$. Let us further assume $E^{\prime}$ has good, super-singular reduction at a prime $\ell\neq p$. Let $K=\overline{\mathbb{Q}}^{\ker(\rho_{E^\prime})}$, show that then the residues field of any prime of $K$ lying above $\ell$ is finite.
Remark: Elkies has proven that infinitely many such primes exist.