Let $\alpha=\sqrt[5]{2} \in \mathbb{R}$ and $\xi=e^{\large \frac{2 \pi i}{5}}$. Let $K=\mathbb{Q}(\alpha \xi)$. Then choose the correct statements:
$(i)$ There exists a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma (K)=K$ and $\sigma \neq identity$
$(ii)$ There exists a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma(K) \neq K$
$(iii)$ There exists a finite extension $E$ of $\mathbb{Q}$ such that $K \subseteq E$ and $\sigma(K) \subseteq E$ for every field automorphism $\sigma$ of $E$
$(iv)$ For all field automorphism $\sigma$ of $K$, $\sigma (\alpha \xi)=\alpha \xi$.
Answer:
Here $K=\mathbb{Q}(\alpha \xi)$ is the finite extension obtained by adding $5$ roots of $2$, it is one kind of cyclotomic extension which is obtaibed by adding roots of unity to rational field $\mathbb{Q}$, here $\xi$ is the $5^{th}$ roots of unity.
Since for any finite extenstion $E$ of a field $F$, there exists an intermediate field $K$ such that $F \subseteq K \subseteq E$.
Thus $K \subseteq E$ and hence $\sigma(K) \subseteq E$.
So $(3)$ is true.
Also $(1)$ is true.
But how to judge option $(2)$ and $(4)$ ?
Please explain me because I want to learn deeply.
Help me.