I want to show that the extension $\mathbb{R}(X) \supseteq \mathbb{R}\left(X^2+\frac{1}{X^2}\right)$ is Galois, and then find its Galois group $G$ and all subgroups with the corresponding subfields.
Normally when we work with extensions over $\mathbb{Q}$ we find an algebraic element $\alpha \in L \backslash \mathbb{Q}$ and an irrreducible polynomial over $\mathbb{Q}$ such that $p(\alpha)=0$. From there we can prove if the extension is actually a splitting field of a polynomial, find the degree of the extension and its Galois group.
How can I approach the problem in this case? Any tips on where I should start?