Is the normal closure of $L:K$ the same as the normal closure of $L:K(α)$?

Question

Let $K⊆K(α)⊆L⊆ℂ$. Is the normal closure of $L:K$ the same as the normal closure of $L:K(α)$?

If so, how could it be proved?

and if not, then what would be a counterexample?

I would really appreciate any help/thoughts.

Answer 1

Consider the case of $K=\mathbb{Q}$, $L=\mathbb{Q}(\alpha)$ where $\mathbb{Q}(\alpha)/\mathbb{Q}$ is a non-normal extension.

Now $L=K(\alpha)$, so is $L/K(\alpha)$ normal? Is $L/K$?