Problems in Algebraic Number Theory

Question

if $z$ is an element of $Q(\zeta)$ ,where $\zeta$ is some $k$ th root of unity then $z^{(1/2)}$ is an element of $Q(\zeta^{(1/2)})$ ?

Answer 1

This is false. Let $\zeta = e^{2\pi i/p}$ for any prime $p$. Since the $p$th cyclotomic field is the same as the $2p$th ($\zeta^{1/2} = -\zeta^{(p+1)/2}$), we have $\mathbb{Q}(\zeta) = \mathbb{Q}(\zeta^{1/2})$. Taking any integer $n \not= \pm p$, $\sqrt{n} \notin \mathbb{Q}(\zeta)$ since one of $\mathbb{Q}(\sqrt{p})$ or $\mathbb{Q}(\sqrt{-p})$ is the unique quadratic subfield of $\mathbb{Q}(\zeta)$.