We can now say a bit more about the relationship between quadratic fields and cyclotomic fields.

Let $\omega = e^{2\pi /p}$ for an odd prime $p$. Recall $disc(\omega)={\pm p^{p-2}}$ where the sign is positive if and only if $p=1 (mod 4)$. Using the definition of the discriminant, we have

$|\sigma_i(\omega^j)| = p^{(p-3)/ 2}\sqrt{\pm p}$

where the $\sigma_i$ are the embeddings of $\mathbb{Q}[\omega]$ in $\mathbb{C}$. But each embedding simply maps each $\omega^i$ to some other $\omega^j$, thus we may compute $\sqrt{\pm p}$ using field operations on the powers of $\omega$. In other words, $\sqrt{\pm p} \in \mathbb{Q}[\omega]$, with the sign positive if and only if $p = 1 (mod 4)$.

For example, for $p = 3$ the above equation becomes

${| \array { 1 & \omega \\ 1 & \omega^2 } |} = \sqrt{-3}$

which can be rewritten $\sqrt{-3} = 2\omega + 1$.

Similarly for $p = 5$ we obtain $\sqrt{5} = \omega^2 - \omega^4 + \omega^3 - \omega = 1 -2 \omega -2\omega^4$.

The $8$th cyclotomic field contains $\sqrt{2}$ because in this case we have $\omega = \sqrt{2}/2 + i\sqrt{2}/2$, and hence $\sqrt{2} = \omega + \omega^{-1}$.

If the $q$th cyclotomic field contains $\mathbb{Q}[\sqrt{p}]$, the $4q$th cyclotomic field contains $\mathbb{Q}[\sqrt{- p}]$ because it must contain the fourth root of unity $i$ along with $\sqrt{p}$.

Now consider any squarefree $m = p_1 ... p_r$. For each $p_i$ take the cyclotomic field containing $\sqrt{p}$. Then take the smallest cyclotomic field $K$ containing all these fields. Then $K$ contains $\mathbb{Q}[\sqrt{m}]$. Set $d = disc(\mathbb{A}\cap\mathbb{Q}[\sqrt{m}])$. It can be easily verified that the desired $K$ is in fact the $d$th cyclotomic field.

Kronecker and Weber proved that every abelian extension of $\mathbb{Q}$ (normal with abelian Galois group) is contained in a cyclotomic field. Hilbert and others studied abelian extensions of general number fields, and their results are known as class field theory.