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<j\leq q-1}(\zeta_{q}^{i}-\zeta_{q}% ^{j})^{2}

Taking square roots we obtain

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 $q-1\equiv 0,1\ \operatorname{mod}\ 4$ ). ∎

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.

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.

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 $1<n<f,p^{n}\neq 1\ \operatorname{mod}\ q$ ).

References

1 Daniel A.Marcus, Number Fields. Springer, New York.

Title	prime ideal decomposition in cyclotomic extensions of $\mathbb{Q}$
Canonical name	PrimeIdealDecompositionInCyclotomicExtensionsOfmathbbQ
Date of creation	2013-03-22 13:53:49
Last modified on	2013-03-22 13:53:49
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11R18
Related topic	PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ
Related topic	CalculatingTheSplittingOfPrimes
Related topic	KroneckerWeberTheorem
Related topic	ExamplesOfPrimeIdealDecompositionInNumberFields
Related topic	SplittingAndRamificationInNumberFieldsAndGaloisExtensions

prime ideal decomposition in cyclotomic extensions of ℚ

Proposition 1.

Proof.

Theorem 1.

References

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