Let $F=\mathbb{C}(x, y)$ be the function field in two variables $x, y$. Let $n \geqslant 1$, and $K=\mathbb{C}\left(x^{n}+y^{n}, x y\right)$ be the subfield of $F$ of all rational functions in $x^{n}+y^{n}$ and $x y .$

(i) Let $K^{\prime}=K\left(x^{n}\right)$, which is a subfield of $F$. Show that $K^{\prime} / K$ is a quadratic extension.

(ii) Show that $F / K^{\prime}$ is cyclic of order $n$, and $F / K$ is Galois. Determine the Galois $\operatorname{group} \operatorname{Gal}(F / K)$.