Let $F$ be a field and $f(x) \in F[x]$ a nonconstant polynomial. If $K $ is a splitting field of $f(x)$ over $F$ and $L$ is any extension field of $F$, suppose that $\alpha : K \rightarrow L $ is a homomorphism satisfying $\alpha (c) = c$ for all $c \in F$. If $a \in K$ is a root of $f(x)$, show that $\alpha(a)$ is also a root of $f(x)$.
In the case that $a\in F$, it's esay to see that the statement is true, since $\alpha(a)=a$. But what about the case where $a\in K\setminus F$?
I have been trying to understand the result by means of examples, but so far I have nothing. Could you help me with some hint, please?