Let $k$ be a field with 8 elements and $K$ a field with 64 elements. How many ring homomorphisms are there from $k$ to $K$?
This question was asked here but it has no proper answers. As much as I have understood any field homomorphism from $k$ to $K$ will be a $k$-automorphism that fixes $\mathbb{F}_2$. Also, any finite extension of a finite field is Galois. Hence, the number of homomorphisms equals $|Gal(k/\mathbb{F}_2)|$ i.e., $3$.