Let $k$ be a field, $f(x)\in k[x]$, and let $F$ be the splitting field of $f(x)$ over $k$. Let $k\subseteq K$ be an extension such that $f(x)$ splits as a product of linear factors over $K$. Prove that there is a homomorphism $F\to K$ extending the identity on $k$.
This is a question from Aluffi's Algebra Chapter 0.
It looks to me as if here, $K$ is just the algebraic closure of $F$. Thus, a homomorphism would just be the inclusion. Is this all there is to it or am I missing something?