Roots of unity in quartic fields It's a well known fact that the roots of unity of the ring of integers of $\mathbb{Q}(\sqrt d)$ where $d\in \mathbb{Z}$ is squarefree, is $\{\pm 1, \pm i\}$ when $d=-1$; $\{\pm 1, \pm \omega, \pm \omega^2\}$, $\omega=e^{\frac{2\pi i}{3}}$ when $d=-3$, and $\{\pm 1\}$ otherwise.
What is known of the roots of unity in quartic number fields?

The possible $p$-primary $n$-th roots of unity $\zeta_n$ are


*

*$\zeta_2$ is in a degree-1 extension $\mathbb{Q}$

*$\zeta_4$ is in a degree-2 extension $\mathbb{Q}(i)$

*$\zeta_8$ is in a degree-4 extension $\mathbb{Q}(i, \sqrt{2})$

*$\zeta_3$ is in a degree-2 extension $\mathbb{Q}(\sqrt{-3})$

*$\zeta_5$ is in a degree-4 extension


You can mix and match these as you like to fit them into a degree $4$ extension.
e.g. $\mathbb{Q}(\zeta_{12}) = \mathbb{Q}(\zeta_3, \zeta_4) = \mathbb{Q}(i, \sqrt{3})$ is a degree $2 \cdot 2 = 4$ of $\mathbb{Q}$, built from two independent degree-2 extensions.
Every other $p$-primary root of unity lies in a higher degree extension. e.g.


*

*$\zeta_{16}$ is in a degree-8 extension

*$\zeta_{9}$ is in a degree-6 extension

*$\zeta_{25}$ is in a degree-20 extension 

*$\zeta_p$ is in a degree $p-1$ extension for $p > 5$


(IIRC, the general formula is $[ \mathbb{Q}(\zeta_n) : \mathbb{Q} ] = \varphi(n)$, if you want to generalize to other situations)
