Kronecker-Weber theorem

The following theorem classifies the possible extensionsMathworldPlanetmath of .

Theorem 1 (Kronecker-Weber Theorem).

Let L/Q be a finite extension, then L is contained in a cyclotomic extension, i.e. there is a root of unityMathworldPlanetmath ζ such that LQ(ζ).

In a similar fashion to this result, the theory of elliptic curvesMathworldPlanetmath with complex multiplicationMathworldPlanetmath provides a classification of extensions of quadratic imaginary number fields:

Theorem 2.

Let K be a quadratic imaginary number field with ring of integersMathworldPlanetmath OK. Let E be an elliptic curve with complex multiplication by OK and let j(E) be the j-invariant of E. Then:

  1. 1.

    K(j(E)) is the Hilbert class fieldMathworldPlanetmath of K.

  2. 2.

    If j(E)0,1728 then the maximal extension of K is given by:


    where h(Etorsion) is the set of x-coordinates of all the torsion points of E.

Note: The map h:E is called a Weber function for E. We can define a Weber function for the cases j(E)=0,1728 so the theorem holds true for those two cases as well. Assume E:y2=x3+Ax+B, then:

h(P)={x(P), if j(E)0,1728;x2(P), if j(E)=1728;x3(P), if j(E)=0.


Title Kronecker-Weber theorem
Canonical name KroneckerWeberTheorem
Date of creation 2013-03-22 13:52:41
Last modified on 2013-03-22 13:52:41
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 11R20
Classification msc 11R37
Classification msc 11R18
Related topic ComplexMultiplication
Related topic AbelianExtension
Related topic PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ
Related topic NumberField
Related topic CyclotomicExtension
Related topic ArithmeticOfEllipticCurves
Defines abelian extensions of quadratic imaginary number fields
Defines Weber function