I've been solving problems from my Galois Theory course, and I don't know how to approach this one. It says:
Given $K/F$ field extension, and let $E,L$ be in-between fields ($F\subseteq E\subseteq K$ and $F\subseteq L\subseteq K$). Prove that, if $L/F$ is a finite Galois extension, the $EL/E$ is also a finite Galois extension. Also prove $\operatorname{Gal}(EL/E)\cong\operatorname{Gal}(L/E\cap L)$.
I don't really know how to approach this (essentialy the first part). I've tried to direct prove by double content that $$ \operatorname{Fix}(\operatorname{Gal}(EL/E))\subseteq E, \quad E\subseteq\operatorname{Fix}(\operatorname{Gal}(EL/E)), $$ while assuming $[L:F]<\infty$ and $\operatorname{Fix}(\operatorname{Gal}(L/F))=F$, but I'm a bit lost. For the last part, I guess it will have to do with Second Theorem of isomorphism (but I haven't developed it yet since first I wanna understand the first part of the problem).
I thought it would be easy to find online (propably in this forum), and tried SearchOnMath but I found nothing, so I apologise if this is a duplicate. Any help or hint will be appreciated, thanks in advance.
Since $L/F$ is a finite Galois extension the minimal polynomial for every $\alpha\in L$ is separable and splits completely in $L$. Let $p(x)=\min_F(\alpha)\in F[x]$. Now, if $f(x)=\min_E(\alpha)\in E[x]$ then $f(x)\mid p(x)$ viewing $p(x)$ as polynomial over $E$ (which is possible since $F\subseteq E$). Hence $f(x)$ is separable as well. Moreover, $EL$ is normal as the splitting field of the family of polynomials $\{\min_F(\alpha)\}_{\alpha\in L}$. So $EL/E$ is Galois.
For the second part consider the following homomorphism $$ \varphi\colon\operatorname{Gal}(EL/E)\to\operatorname{Gal}(L/E\cap L),\,\sigma\mapsto\sigma|_L\,. $$ One checks immediately that this is well-defined and injective. Surjectivity follows from a careful analysis what the elements in the two Galois groups actually do.