Number Fields

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 $\mathrm{disc}(\omega )=\pm p^{p-2}$ where the sign is positive if and only if $p=1(\mathrm{mod}4)$. Using the definition of the discriminant, we have

$$\mid {\sigma }_i({\omega }^j)\mid =p^{(p-3)/2}\sqrt{\pm p}$$

where the ${\sigma }_i$ are the embeddings of $\mathbb{Q}[\omega ]$ in $ℂ$. 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 $p=1(\mathrm{mod}4)$.

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

$$\mid \begin{array}{cc}1& \omega \\ 1& {\omega }^2\end{array}\mid =\sqrt{-3}$$

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 $d=\mathrm{disc}(𝔸\cap \mathbb{Q}[\sqrt{m}])$. 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 $mathbQ$ (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.