Let $E/K$ and $L/K$ be finite extensions contained in some common field. if $E/K$ is Galois, prove that $EF/L$ and $E/E\cap F$ are also Galois.
It's easy to see that these two field extensions are finite. Then we have to check that these two are normal and separable. For $E/E \cap F$, we can build an extension tower and normality and separability follows from the fact that $E/K$ is. But I have no idea about how to deal with $EF/L$.
Here is a sketch:
(i) Remember, $E/K$ is Galois if and only if $E$ is the splitting field over $K$ for some polynomial $f(X) \in K[X]$ with no repeated roots in $E$. Let us denote the roots of $f(X)$ in $E$ by $\alpha_1, \dots, \alpha_n$. Try to show that $E = K(\alpha_1, \dots, \alpha_n)$.
(ii) By definition, $EF$ is the smallest subfield of the big "common field" that contains both $E$ and $F$. Try to deduce that $EF = F(\alpha_1, \dots, \alpha_n)$.
(iii) Now argue that $EF$ is the splitting field over $F$ of the polynomial $f(X)$, viewed as a polynomial in $F[X]$. $f(X)$ has distinct roots in $EF$, since its roots are simply the original roots $\alpha_1, \dots, \alpha_n$. Hence $EF / F$ is a Galois extension.