$\xi =e^{2*\pi/pi }$ , proof that is only two quadratic extention to $\Bbb Q $ that contained in $Q(\xi_p)$
Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$