I want to answer the following question: Given field extensions $K \subset M \subset L $, when is the case that $M:K$ is normal and $L:M$ is normal implies $L:K$ is normal? In my algebra class we have only considered subfields of $\mathbb{C}$ but I would appreciate an answer involving general fields as well. For example, I try starting with a normal extension $M:K$ and then adjoining elements to $M$ to find the normal $L$, but it seems that $L$ always becomes "too big" in the sense that it is not the splitting field of a polynomial over $M$ because it contains the algebraic element from $M$.
Ex: $\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt{2},\zeta_3, \sqrt[3]{2})$, where $\zeta_3$ is the primitive third root of unity. Certainly $\mathbb{Q}(\sqrt{2}): \mathbb{Q}$ is normal, but $\mathbb{Q}(\sqrt{2},\zeta_3, \sqrt[3]{2}) : \mathbb{Q}(\sqrt{2})$ is not (unless I'm not seeing the polynomial over $\mathbb{Q}(\sqrt{2})$ for which $\mathbb{Q}(\sqrt{2},\zeta_3, \sqrt[3]{2})$ is the splitting field of) since the splitting field of $p(t)=x^3-2$ is $\mathbb{Q}(\zeta_3, \sqrt[3]{2})$ (this is what I mean by $L$ is "too big").
So to be clear, I would like to know how to construct our $L$ (such that $L:M$ is normal) and, in general, under what conditions this $L$ is normal over $K$.