Statements - Galois extension

Question

Let $E/F$ be Galois extension.

I want to show the following:

• $F\leq K_1\leq K_2\leq E\Rightarrow \mathcal{G}(E/K_1)\geq \mathcal{G}(E/K_2)$
• $H_1\leq H_2\leq \mathcal{G}(E/F)\Rightarrow \mathcal{F}(H_1)\geq \mathcal{F}(H_2)$

Could you give me some hints how we could show these relations?

Answer 1

Hint for the first one (and then deduce for the other one, use Galois corresponde or whatever, etc.):

$$\sigma\in\text{Gal}(E/K_2)\implies \sigma(k_2)=k_2\;,\;\forall\;k_2\in K_2$$

but $\;K_1\le K_2\implies\;$ the above is true in $\;K_1\;$ ...