Let $F$ be a field and let $K$ be the splitting field of $F[x]$. Then $K$ is algebraically closed. This is a theorem we proved yesterday in my algebra class. The strategy was to take a polynomial $f(x) \in K[x]$ and look at the intermediate field produced by adjoining the coefficients of $f(x)$ to $F$. I am confused as to how this is necessary however. What is wrong with the following proof?
Let $f(x) \in L[x]$. Now $f(x)$ has a splitting field $L/K$, so there exists some $\alpha \in L$ such that $f(\alpha) = 0$. Since $L/K$ and $K/F$ are both splitting fields, they are both algebraic extensions, hence $L/F$ is algebraic. Therefore $\alpha$ has a minimal polynomial $m_{\alpha}(x) \in F[x]$. Since $K$ is the splitting field of $F[x]$, $m_{\alpha}(x)$ splits over $K$, hence $\alpha \in K$. Therefore $f(x)$ has a root in $K$, so $K$ is algebraically closed. Thanks!