Kronecker-Weber theorem

The following theorem classifies the possible http://planetmath.org/node/AbelianExtensionabelian extensions of $\mathbb{Q}$ .

Theorem 1 (Kronecker-Weber Theorem).

Let $L/\mathbb{Q}$ be a finite http://planetmath.org/node/AbelianExtensionabelian extension, then $L$ is contained in a cyclotomic extension, i.e. there is a root of unity $\zeta$ such that $L\subseteq\mathbb{Q}(\zeta)$ .

In a similar fashion to this result, the theory of elliptic curves with complex multiplication provides a classification of http://planetmath.org/node/AbelianExtensionabelian extensions of quadratic imaginary number fields:

Theorem 2.

Let $K$ be a quadratic imaginary number field with ring of integers $\mathcal{O}_{K}$ . Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K}$ and let $j(E)$ be the $j$ -invariant of $E$ . Then:

1.

$K(j(E))$ is the Hilbert class field of $K$ .

If $j(E)\neq 0,1728$ then the maximal http://planetmath.org/node/AbelianExtensionabelian extension of $K$ is given by:

$K^{ab}=K(j(E),h(E_{\operatorname{torsion}}))$

where $h(E_{\operatorname{torsion}})$ is the set of $x$ -coordinates of all the torsion points of $E$ .

Note: The map $h\colon E\to\mathbb{C}$ 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 $E\colon y^{2}=x^{3}+Ax+B$ , then:

$h(P)=\begin{cases}x(P),\text{ if $j(E)\neq 0,1728$};\\ x^{2}(P),\text{ if $j(E)=1728$};\\ x^{3}(P),\text{ if $j(E)=0$}.\end{cases}$

References

1 S. Lang, Algebraic Number Theory, Springer-Verlag, New York.
2 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York.

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