(i) Let $K \leqslant \mathbb{C}$ be a field and $L \leqslant \mathbb{C}$ a finite normal extension of $K$. If $H$ is a finite subgroup of order $m$ in the Galois group $G(L \mid K)$, show that $L$ is a normal extension of the $H$-invariant subfield $I(H)$ of degree $m$ and that $G(L \mid I(H))=H$. [You may assume the theorem of the primitive element.]

(ii) Show that the splitting field over $\mathbb{Q}$ of the polynomial $x^{4}+2$ is $\mathbb{Q}[\sqrt[4]{2}, i]$ and deduce that its Galois group has order 8. Exhibit a subgroup of order 4 of the Galois group, and determine the corresponding invariant subfield.