prime ideal decomposition in cyclotomic extensions of

Let q be a prime greater than 2, let ζq=e2πi/q and write L=(ζq) for the cyclotomic extension. The ring of integersMathworldPlanetmath of L is 𝒪L=[ζq]. The discriminantPlanetmathPlanetmathPlanetmath of L/ is:


and it is + exactly when q-10,1mod 4.

Proposition 1.

±q(ζq), with + exactly when q-10,1mod 4.


It can be proved that:


Taking square roots we obtain


Hence the result holds (and the sign depends on whether q-10,1mod 4). ∎

Let K=(±q) with the corresponding sign. Thus, by the proposition we have a tower of fields: \xymatrix&L=(ζq)\ar@-[d]&K\ar@-[d]&

For a prime idealMathworldPlanetmathPlanetmath p the decomposition in the quadratic extension K/ is well-known (see entry). The next theorem characterizes the decomposition in the extension L/:

Theorem 1.

Let pZ be a prime.

  1. 1.

    If p=q, q𝒪L=(1-ζq)q-1. In other words, the prime q is totally ramified in L.

  2. 2.

    If pq then p splits into (q-1)/f distinct primes in 𝒪L, where f is the order of pmodq (i.e. pf1modq, and for all 1<n<f,pn1modq).


Title prime ideal decomposition in cyclotomic extensions of
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