Kronecker-Weber theorem

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

Theorem 1 (Kronecker-Weber Theorem).

Let $L\mathrm{/}\mathrm{Q}$ be a finite http://planetmath.org/node/AbelianExtensionabelian extension, then $L$ is containedin 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 curveswith complex multiplication provides a classification of http://planetmath.org/node/AbelianExtensionabelianextensions of quadratic imaginary number fields:

Theorem 2.

Let $K$ be a quadratic imaginary number field with ring ofintegers ${\mathrm{O}}_K$. Let $E$ be an elliptic curve withcomplex 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 Weberfunction for $E$. We can define a Weber function for the cases$j(E)=0,1728$ so the theorem holds true for those two cases aswell. Assume $E:y^2=x^3+Ax+B$, then:

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