Determine the quadratic number fields $\mathbb Q[\sqrt{d}]$ that contain a primitive $n$-th root of unity, for some integer $n$.

Question

Determine the quadratic number fields $\mathbb Q[\sqrt{d}]$ that contain a primitive $n$-th root of unity, for some integer $n$.

I've determined using the cyclotomic polynomial that for $n=2,3,6$, the primitive $n$-th root of unity will have a degree $2$ minimal polynomial. But how do I determine which quadratic number fields $\mathbb Q[\sqrt{d}]$ will contain a primitive $n$-th root of unity?

Answer 1

A primitive $n$-th root of unity has degree $\varphi(n)$.

If a quadratic number fields contain a primitive $n$-th root of unity, then $\varphi(n)\le2$. This happens iff $n \in \{1,2,3,4,6\}$.