(i) Prove that a finite solvable extension $K \subseteq L$ of fields of characteristic zero is a radical extension.

(ii) Let $x_{1}, \ldots, x_{7}$ be variables, $L=\mathbb{Q}\left(x_{1}, \ldots, x_{7}\right)$, and $K=\mathbb{Q}\left(e_{1}, \ldots, e_{7}\right)$ where $e_{i}$ are the elementary symmetric polynomials in the variables $x_{i}$. Is there an element $\alpha \in L$ such that $\alpha^{2} \in K$ but $\alpha \notin K$ ? Justify your answer.

(iii) Find an example of a field extension $K \subseteq L$ of degree two such that $L \neq K(\sqrt{\alpha})$ for any $\alpha \in K$. Give an example of a field which has no extension containing an $11 t h$ primitive root of unity.