Suppose that $C$ is an algebraic closure of $F$, $f\in F[x]$ irreducible and $a,b\in C$ roots of $f$.
We have the field extension $F\leq F[x]/\langle f\rangle$. That means that $F\rightarrow F[x]/\langle f\rangle$ is an homomoprhism, right?
Does it hold that $C$ is also an algebraic closure of $F[x]/\langle f\rangle$ ?