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

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

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

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:

 $D_{L/\mathbb{Q}}=\pm q^{q-2}=\prod_{1\leq i

Taking square roots we obtain

 $q^{\frac{q-3}{2}}\sqrt{\pm q}=\prod_{1\leq i

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

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 http://planetmath.org/encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.htmlthis entry). The next theorem characterizes the decomposition in the extension $L/\mathbb{Q}$:

###### Theorem 1.

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

1. 1.

If $p=q$, $q\mathcal{O}_{L}=\left(1-\zeta_{q}\right)^{q-1}$. In other words, the prime $q$ is totally ramified in $L$.

2. 2.

If $p\neq q$ then $p\mathbb{Z}$ splits into $(q-1)/f$ distinct primes in $\mathcal{O}_{L}$, where $f$ is the order of $p\ \operatorname{mod}\ q$ (i.e. $p^{f}\equiv 1\ \operatorname{mod}\ q$, and for all $1).

## References

• 1 Daniel A.Marcus, . Springer, New York.
Title prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$ PrimeIdealDecompositionInCyclotomicExtensionsOfmathbbQ 2013-03-22 13:53:49 2013-03-22 13:53:49 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 11R18 PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ CalculatingTheSplittingOfPrimes KroneckerWeberTheorem ExamplesOfPrimeIdealDecompositionInNumberFields SplittingAndRamificationInNumberFieldsAndGaloisExtensions