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?

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Let $F$ be a quartic extension of $\mathbb{Q}$. Let $\mathbb{Q}_{k} = \mathbb{Q}(e^{\frac{2\pi i}{k}})$.

Let $\mathbb{Q}_d$ be the maximal Cyclotomic field contained in $F$ (note that $\mathbb{Q}_{d_1}, \mathbb{Q}_{d_2} \subseteq F \implies \mathbb{Q}_{lcm(d_1,d_2)} \subseteq F$, and that if a primitive root of unity of order $k$ is contained in $F$ we have $\mathbb{Q}_{k} \subseteq F$. Thus, $\mathbb{Q}_{d}$ is well-defined and contains all the roots of unity contained in $F$).

Note that $\mathbb{Q}_{d} =\mathbb{Q}_{2d}$ when $d$ is odd, so we can assume the either $d$ is odd or divisible by $4$.

We have the following tower of extensions: $\mathbb{Q} \subseteq \mathbb{Q}_{d} \subseteq F$. This implies that $\varphi(d) = [\mathbb{Q}_{d} : \mathbb{Q}] | [F:\mathbb{Q}] = 4$. The relation $\varphi(d) | 4$ implies, together with the assumption on $d$, that $d | 5$ or $d|12$ or $d|8$.

So, the distinct options:

  1. $d=5, 8$ or $12$, in which case $\phi(d)=4$ and $F=\mathbb{Q}_{d}$.

  2. $d =3$ or $d=4$, in which case $\phi(d)=2$ and $F$ is a quadratic extension of $\mathbb{Q}_{d}$: $F=\mathbb{Q}(i,\sqrt{a+bi})$ or $F=\mathbb{Q}(\omega,\sqrt{a+b\omega})$

  3. $d=1$, in which case $F$ doesn’t contain roots of unity other than $\pm 1$.

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)