Extension degree of galois closure of $\mathbb{R}(X^4 -\frac{1}{X^4 })$ Let rational function field $K=\mathbb{R}(X)$ , $F= \mathbb{R}(X^4 -\frac{1}{X^4 })$ and $L$ be the Galois closure of $K/F$.
I can't write L and don't understand $[L:F]$ .
Please find $[L:F]$.

$K = F(X)$, so the Galois closure of $K$ over $F$ is the splitting field of the minimum polynomial of $X$ over $F$.
The minimum polynomial of $X$ over $F$ is
$$t^8 - (X^4 - \frac 1 {X^4})t^4 - 1.$$
This has roots $\pm X, \pm iX, \pm \frac {\zeta_8} X, \pm \frac {i\zeta_8} X$. So the Galois closure is $\mathbb C(X)$.
Does this help?
