PlanetMath: prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$

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prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$

(Theorem)

Let $q\in\mathbb{Z}$ be a prime greater than , let $\zeta_q=e^{2\pi i/q}$ and write $L=\mathbb{Q}(\zeta_q)$ for the cyclotomic extension. The ring of integers of is $\mathcal{O}_L=\mathbb{Z}[\zeta_q]$ . The discriminant of $L/\mathbb{Q}$ is:

$\displaystyle D_{L/\mathbb{Q}}=\pm q^{q-2}$

and it is

exactly when $q-1\equiv 0,1\ \operatorname{mod}\ 4$ .

Proposition 1 $\sqrt{\pm q}\in \mathbb{Q}(\zeta_q)$ , with exactly when $q-1\equiv 0,1\ \operatorname{mod}\ 4$ .

Proof. It can be proved that:

$\displaystyle D_{L/\mathbb{Q}}=\pm q^{q-2}=\prod_{1\leq i < j \leq q-1}(\zeta_q^i-\zeta_q^j)^2$

Taking square roots we obtain

$\displaystyle q^{\frac{q-3}{2}}\sqrt{\pm q}=\prod_{1\leq i < j \leq q-1}(\zeta_q^i-\zeta_q^j)\in \mathbb{Q}(\zeta_q)$

Hence the result holds (and the sign depends on whether $q-1\equiv 0,1 \operatorname{mod}\ 4$ ). $\qedsymbol$

Let $K=\mathbb{Q}(\sqrt{\pm q})$ with the corresponding sign. Thus, by the proposition we have a tower of fields: $\xymatrix{ & L=\mathbb{Q}(\zeta_q) \ar@{-}[d]\ & K \ar@{-}[d]\ & \mathbb{Q}}$

For a prime ideal $p\mathbb{Z}$ the decomposition in the quadratic extension $K/\mathbb{Q}$ is well-known (see this entry). The next theorem characterizes the decomposition in the extension $L/\mathbb{Q}$ :

Theorem 1 Let $p\in\mathbb{Z}$ be a prime.

If , $q\mathcal{O}_L=\left(1-\zeta_q\right)^{q-1}$ . In other words, the prime is totally ramified in .
If $p\neq q$ then $p\mathbb{Z}$ splits into distinct primes in $\mathcal{O}_L$ , where is the order of $p \operatorname{mod}\ q$ (i.e. $p^f\equiv 1\ \operatorname{mod}\ q$ , and for all $1< n<f, p^n\neq 1\ \operatorname{mod}\ q$ ).