Let $F \subset G_i \subset E, (i=1,2)$ where $E$ is normal and finite dimensional over $F$. Assume $E$ is separable over $G_1$ and $G_2$. Prove $E$ is separable over $G_1 \cap G_2$
There is a hint to use this theorem that states that a finite dimensional field extension is Galois iff it is normal and separable. I think that this means we need to show that $G_i \subset E$ are Galois, which in this case is equivalent to showing that the extensions are normal. I have a proposition that states that any intermediate field extension is normal so these extensions are normal and therefore Galois since we are given separable. I do not see how $G_1, G_2 \subset E$ being Galois implies that their intersection is separable. Is the intersection of two Galois extensions Galois or am I supposed to just use the definition of separable (in which case, why did I even find they are Galois), to show that $E$ is separable over $G_1 \cap G_2$?