characterizing CM-fields using Dirichlet’s unit theorem

If K is a number fieldMathworldPlanetmath, 𝒪K is the ring of algebraic integers in K, and 𝒪K is the (multiplicative) group of units in 𝒪K. Dirichlet’s unit theorem gives the structureMathworldPlanetmath of the unit group. We can use that theorem to characterize CM-fields:

Theorem 1.

Let QFK be nontrivial extensionsPlanetmathPlanetmathPlanetmath of number fields. Then K is a CM-field, with F its totally real subfieldMathworldPlanetmath, if and only if OK/OF is finite.

We use the notation of the article on Dirichlet’s unit theorem, where r (and rF,rK) is used to count real embeddings and s (as well as sF,sK) to count complex embeddings, and we write μ(F) or μ(K) for the group of roots of unityMathworldPlanetmath in 𝒪F or 𝒪K.

Write n=[F:],m=[K:F]>1.

(): If K/F is CM, then since F is totally real, rF=n,sF=0. Hence by Dirichlet’s unit theorem, 𝒪Fμ(F)×n-1. Since K/F is a complex quadratic extension, [K:]=2n and all its embeddingsPlanetmathPlanetmath are complex. Thus rK=0, 2sK=2n. Hence 𝒪Kμ(K)×n-1 as well. Clearly 𝒪F𝒪K, and since they have the same rank (, their quotient is torsion and thus finite.

(): Since 𝒪K/𝒪F is finite, the ranks of these groups are equal and thus rF+sF=rK+sK again by Dirichlet’s unit theorem.


rK+2sK =mn=m(rF+2sF) (1)
rK+sK =rF+sF; (2)

subtracting (2) from (1), we get

sK=(m-1)(rF+2sF)+sF(m-1)n (3)

and thus mn=rK+2sKrK+2(m-1)n so that 0rKn(2-m). Thus m2, and since K is a nontrivial extension, we must have m=2 so that K/F is quadratic and rK=0 (since n(2-m)=0).

Finally, by (3), we then have sK=rF+3sF; (2) says that sK=rF+sF, and thus sF=0. It follows that F is totally real and, since rK=0, K must be an imaginary quadratic extension of F.

Title characterizing CM-fields using Dirichlet’s unit theorem
Canonical name CharacterizingCMfieldsUsingDirichletsUnitTheorem
Date of creation 2013-03-22 17:57:26
Last modified on 2013-03-22 17:57:26
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 11R04
Classification msc 11R27
Classification msc 12D99