Find the fixed field of the following subgroup? I am trying to understand some concepts via random exercises I found from past papers but this particular one, I am not sure even where to start.
There aren't any solutions for the paper so would someone be kind enough to provide me with a standard solution, so I can study the flow and try figuring out the ideas myself? I might comment and ask if simply I cannot digest a part of it

Let $\omega = e^{\frac{2 \pi i}{5}}$. So then the $\mathbb{Q}(\omega)$ consists of all elements of the form $p+q \omega+r\omega^2+s\omega^3+t\omega^4 $. Let the $\mathbb{Q}$-automorphisms be
$$\alpha_1: p+q \omega+r\omega^2+s\omega^3+t\omega^4 \rightarrow p+q \omega+r\omega^2+s\omega^3+t\omega^4 $$
$$\alpha_2: p+q \omega+r\omega^2+s\omega^3+t\omega^4 \rightarrow p+s \omega+q\omega^2+t\omega^3+r\omega^4 $$
$$\alpha_3: p+q \omega+r\omega^2+s\omega^3+t\omega^4  \rightarrow p+r \omega+t\omega^2+q\omega^3+s\omega^4 $$
$$\alpha_4: p+q \omega+r\omega^2+s\omega^3+t\omega^4  \rightarrow p+t \omega+s\omega^2+r\omega^3+q\omega^4 $$
Take the subgroup $\{\alpha_1, \alpha_4\}$ and find its fixed field.

So I am starting with the Galois group $\{\alpha_1, \alpha_2, \alpha_3, \alpha_4\}$, and considering a particular subgroup. I don't have a method to proceed, so it would be very much appreciated if I can find a way to answer it, thank you in advance!

Let $E/F$ be a finite Galois extension fields of characteristic zero.  Let $n$ be the degree of the extension, $G$ the Galois group.  Suppose that $\{x_1,\ldots,x_n\}$ is basis for $E$ over $F$.  Let $H$ be a subgroup of $G$, and consider the elements $y_i = \sum_{h\in H} h\cdot x_i$.  Show that $\{y_1,\ldots,y_n\}$ span the fixed field of $H$.
