index of the group of cyclotomic units in the full unit group

Let Kn=(ζpn) where ζpn is a primitive pnth root of unityMathworldPlanetmath, let hn be the class numberMathworldPlanetmathPlanetmath of Kn and let 𝒪n=𝒪Kn be the ring of integersMathworldPlanetmath in Kn. Let En=𝒪n× be the group of units in Kn. The cyclotomic units are a subgroupMathworldPlanetmathPlanetmath Cn of En which satisfy:

  • The elements of Cn are defined analytically.

  • The subgroup Cn is of finite index in En. Furthermore, the index is hn+: Let En+ be the group of units in Kn+ and let Cn+=CnEn+. Then [En+:Cn+]=hn+. Moreover, it can be shown that [En:Cn]=[En+:Cn+] because En=μpnEn+ (this is exercise 8.5 in [1]).

  • The subgroups Cn behave “well” in towers. More precisely, the norm of Cn+1 down to Kn is Cn. This follows from the fact that the norm of ζpn+1 down to Kn is ζpn.

Definition 1.

Let p be prime and let n1. Let ζpn be a primitive pnth root of unity.

  1. 1.

    The cyclotomic unit group Cn+Kn+=(ζpn)+ is the group of units generated by -1 and the units


    with 1<a<pn2 and gcd(a,p)=1.

  2. 2.

    The cyclotomic unit group CnKn=(ζpn) is the group generated by ζpn and the cyclotomic units Cn+ of Kn+.

Remark 1.

Let σa:ζpnζpna be an element of Gal(Kn/). Then:

Remark 2.

Let g be a primitive rootMathworldPlanetmath modulo pn. Let agrmodpn. Then one can rewrite ξa as:


In particular ξg generates Cn+/{±1} as a module over [Gal((ζpn)+/)].

Notice that in order to show that the index of Cn in Kn is finite it suffices to show that the index of Cn+ in Kn+ is finite. Indeed, let [Kn:]=2d. Since Kn is a totally imaginary field and by Dirichlet’s unit theorem the free rank of En is r1+r2-1=d-1. On the other hand, [Kn+:]=d and Kn+ is totally real, thus the free rank of En+ is also d-1. Therefore the free rank of En+ and En are equal. As we claimed before, the index [En+:Cn+] is rather interesting to us.

Theorem 1 ([1],Thm. 8.2).

Let p be a prime and n1. Let hn+ be the class number of Q(ζpn)+. The cyclotomic units Cn+ of Q(ζpn)+ are a subgroup of finite index in the full unit group En+. Furthermore:


In the proof of the previous theorem one calculates the regulatorMathworldPlanetmath of the units ξa in terms of values of Dirichlet L-functions with even charactersPlanetmathPlanetmath. In particular, one calculates:


where in the last equality one uses the properties of Gauss sums and the class number formulaMathworldPlanetmath in terms of Dirichlet L-functions evaluated at s=1. This yields that R({ξa}) in non-zero, therefore the index in En+ is finite and moreover


An immediate consequence of this is that if p divides hn+ then there exists a cyclotomic unit γCn+ such that γ is a pth power in En+ but not in Cn+.


  • 1 L. C. Washington, Introduction to Cyclotomic FieldsMathworldPlanetmath, Second Edition, Springer-Verlag, New York.
Title index of the group of cyclotomic units in the full unit group
Canonical name IndexOfTheGroupOfCyclotomicUnitsInTheFullUnitGroup
Date of creation 2013-03-22 15:42:49
Last modified on 2013-03-22 15:42:49
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 11R18