an exact sequence for ray class groups

Let K be a number fieldMathworldPlanetmath, let π’ͺK be its ring of integersMathworldPlanetmath and let π”ͺ be a modulusMathworldPlanetmathPlanetmathPlanetmath in K, i.e.


where π”ͺ0 is an integral ideal in π’ͺK and π”ͺ∞ is a product of real infinite places (i.e. real archimedeanPlanetmathPlanetmathPlanetmath primes). Let Cl⁑(K) be the ideal class groupPlanetmathPlanetmathPlanetmath of K and let Cl⁑(K,π”ͺ) be the ray class group of K of conductorPlanetmathPlanetmath π”ͺ. Also, define


where |π”ͺ∞| denotes the number of real places in π”ͺ. Finally, let U=π’ͺKΓ— be the unit group of K.


The elements above fit in the following exact sequence:

Example 1.

Let K=β„š. Thus, Cl⁑(β„š) is trivial and U={Β±1}β‰…β„€/2⁒℀. Let π”ͺ=p⁒∞ where p>2 is any prime. Then:


The exact sequence now reads:


Therefore, Cl⁑(β„š,p⁒∞)β‰…(β„€/p⁒℀)Γ—. In fact, as we know, the class field of β„š of conductor π”ͺ=p⁒∞ is the cyclotomic fieldMathworldPlanetmath β„šβ’(ΞΆp) where ΞΆp is any primitive pth root of unityMathworldPlanetmath. Moreover


Finally notice that the ray class group of β„š of conductor π”ͺ=p is simply (β„€/p⁒℀)Γ—/{Β±1} which corresponds to the ray class field β„šβ’(ΞΆp)+=β„šβ’(ΞΆp+ΞΆp-1), the maximal real subfieldMathworldPlanetmath of β„šβ’(ΞΆp).

Title an exact sequence for ray class groups
Canonical name AnExactSequenceForRayClassGroups
Date of creation 2013-03-22 15:42:46
Last modified on 2013-03-22 15:42:46
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Result
Classification msc 11R29
Related topic Modulus
Related topic RayClassField
Related topic ClassNumbersAndDiscriminantsTopicsOnClassGroups