I want to know which is the quartic subfield of the cyclotomic field $\mathbb{Q}(\zeta_p)$ where $p$ is an odd prime? it is $\mathbb{Q}(\sqrt{\varepsilon\sqrt{p}})$ where $\varepsilon$ is the fundamental unit of $\mathbb{Q}(\sqrt p)$? how to prove that?
Also, what are the sufields of a cyclotomic field $\mathbb{Q}(\zeta_n)$ where $n=2^a$?
Cordially