The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi i/3)$.
I'm wondering if this list is complete but I had no idea until I searched MSE, which returns the following question:
Roots of unity in quartic fields
The answer seems to be yes since the OP of that question says that this is a well known fact. But I have no experience in algebraic number theory at all.
Can some one guide me to some references (or give a proof?) about this result?
Let $\zeta$ be a primitive $n$th root of unity. The cyclotomic field $\Bbb Q(\zeta)$ is the splitting field of $x^n-1$. The minimal polynomial of $\zeta$ is the cyclotomic polynomial $\Phi_n(x)$ which has degree $\varphi(n)$, so the field extension $\Bbb Q(\zeta)/\Bbb Q$ has degree $\varphi(n)$. If $\zeta$ is also a quadratic integer then $\Bbb Q(\zeta)/\Bbb Q$ has degree $\le2$ and the only naturals $n$ for which $\varphi(n)\le2$ are $n=1,2,3,4,6$.