## Quadratic 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 $$disc(\omega)={\pm p^{p-2}}$$ 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.

Ben Lynn blynn@cs.stanford.edu 💡