Let $\alpha=i$, $\beta =-i$, $\gamma=\sqrt{2}$ and $\delta=-\sqrt{2}$ and consider the permutations
$$R=\begin{pmatrix}\alpha&\beta&\gamma&\delta\\\beta&\alpha&\gamma&\delta\end{pmatrix},~S=\begin{pmatrix}\alpha&\beta&\gamma&\delta\\\alpha&\beta&\delta&\gamma\end{pmatrix}.$$
I need to prove that the permutations $R$ and $S$ preserve every valid polynomial equation over $\mathbb Q$ relating $\alpha,\beta,\gamma$ and $\delta$.
Some of these valid polynomial equations are
$$\alpha^2+1=0,~\alpha+\beta=0,~\delta^2-2=0,~\gamma+\delta=0,~\alpha\gamma-\beta\delta=0$$
We see that $R$ and $S$ preserve them, but I don't know what is the general form of these polynomials, in order to show that $R$ and $S$ preserve them. Could anyone help me, please?
For convenience, I am going to rewrite things a little bit:
Let $K=\mathbb{Q}(\alpha,\delta)$
Let $R: K\to K$ defined by $\alpha\mapsto -\alpha$ and $\gamma\mapsto \gamma$
Let $S: K\to K$ defined by $\alpha\mapsto \alpha$ and $\gamma\mapsto -\gamma$
Now $G=\langle R, S\rangle $ is a group of order $4$.
You may try to prove that $[K:\mathbb{Q}]=4$ (Not too difficult). Note that this is actually a normal extension and separable extension(characteristic of $K$ is clearly $0$ as it is an extension of $\mathbb{Q}$) It is in fact a Galois extension.
Then, $G=Gal(K/\mathbb{Q})$. If we want to look at all the fixed elements under $G$, you may recall from FTGT that $K^G=\mathbb{Q}$(fixed field of $G$.) So in particular, $p(\alpha,\gamma)\in \mathbb{Q}$ where $p(X,Y)\in \mathbb{Q}[X]$ , it means $R$ and $S$ maps $p(\alpha,\gamma)$ to itself.