Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?
Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$