Verify fundamental theorem of Galois for splitting field of $X^4\pm2\in\mathbb {Q}[X]$

Question

My attempt:

Let $f(X) = X^4 − 2$. Let $K$ be the splitting field of $f(X)$ over $\mathbb Q$. Now we have factorization: $$X^4 − 2 = (X^2− \sqrt2)(X^2+ \sqrt2) = (X − \sqrt[4]{2})(X + \sqrt[4]{2})(X −\sqrt[4]{2}i)(X +\sqrt[4]{2}i)$$

Then $K = \mathbb Q[\sqrt[4]{2},i]$

From here problem $2(ii)$ then I can show that the Galois group is Dihedral group $D_8$ of $8$ elements.

However,

1. The case of $X^4 + 2$ on factorisation goes into $\sqrt i$ which are $\pm {1 \over \sqrt{2}} (1+i)$, and I'm not sure how to deal with that (?)
2. If I go case by case for each subgroup of $D_8$, what shows that there cannot be any other intermediate fields? How exactly do I "verify" the fundamental theorem?