I'm dealing with the question. Let char$K=0$ and $F/K$ be a finite and normal extension. Now, given $g(x)\in K[x]$ and $L$ be the splitting field of $g(x)$ over $F$. Show that $L/K$ is a normal extension.
What I have done is assuming $\alpha_1,\cdots,\alpha_n$ be roots of $g(x)$, then I can write $L=F(\alpha_1,\cdots,\alpha_n)$. However, I'm not sure what I can do next to show that $\forall \alpha\in L$, $m_{\alpha, K}$ splits over $L$ where $m_{\alpha, K}$ is the minimal polynomial of $\alpha$ over K.