Kronecker-Weber theorem

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

Theorem 1 (Kronecker-Weber Theorem).

Let $L\mathrm{/}\mathrm{Q}$ 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 $L\mathrm{\subseteq }\mathrm{Q}\mathit{}\mathrm{(}\zeta \mathrm{)}$.

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 ${\mathrm{O}}_K$. Let $E$ be an elliptic curve with complex multiplication by ${\mathrm{O}}_K$ and let $j\mathit{}\mathrm{(}E\mathrm{)}$ be the $j$-invariant of $E$. Then:

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

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

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

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

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

$$h(P)=\{\begin{array}{cc}x(P),\text{ if }j(E)\ne 0,1728;\hfill & \\ x^2(P),\text{ if }j(E)=1728;\hfill & \\ x^3(P),\text{ if }j(E)=0.\hfill & \end{array}$$

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