Every quadratic number field is contained in a cyclotomic field

Question

Why is every quadratic number field $\mathbb Q(\sqrt{d})$ is contained in a cyclotomic field $\mathbb Q(C_n)$, where $C_n$ is the primitive n-th root of unity?

Answer 1

Write $\zeta_n=\exp(2\pi i/n)$.

When $p\equiv1\pmod 4$ is prime, then $\sqrt p=\sum_{k=0}^{p-1}\zeta_p^{k^2}$ (quadratic Gauss sum).

When $p\equiv3\pmod 4$ is prime, then $i\sqrt p=\sum_{k=0}^{p-1}\zeta_p^{k^2}$ (quadratic Gauss sum).

$i=\zeta_4$.

$\sqrt 2=\zeta_8+\zeta_8^7$.

So all these square roots are in some cyclotomic field. But every $\sqrt d$ for $d\in\Bbb Z$ is a product of a bunch of these, so also must lie in some cyclotomic field.