Let $G/H$ be a Galois extension and let $N_1$ and $N_2$ be subfields between $G$ and $H$

Question

Show that: $Gal(G/(N_1N_2)) = Gal(G/N_1) \cap Gal(G/N_2)$

Um, one direction seems pretty obvious by definition: I believe it's that $Gal(G/N_1) \cap Gal(G/N_2) \subseteq Gal(G/N_1N_2)$

Now I have no idea how to that: $Gal(G/N_1N_2)\subseteq Gal(G/N_1) \cap Gal(G/N_2)$. Any ideas how to show this? I'm guessing it has to start with using the fundamental theorem of galois theory but I don't know where to start.

Answer 1

$$\sigma\in\text{Gal}\left(G/N_1N_2\right)\implies \sigma x=x\;,\;\;\forall\,x\in N_1N_2$$

But $\;N_i\subset N_1N_2\;,\;\;i=1,2\;$ , so $\;\sigma n_i=n_i\;,\;\;\forall n_1\in N_i\;,\;\;i=1,2\;$