Kronecker-Weber theoremKroneckerWeberTheorem2003-08-19 10:58:392006-10-02 09:54:05Theoremabelian extensionabelian extensions of quadratic imaginary number fieldsWeber function% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Nats}{\mathbb{N}}
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\newcommand{\Rats}{\mathbb{Q}}The following theorem classifies the possible \PMlinkid{abelian extensions}{AbelianExtension}
of $\Rats$.
\begin{thm}[Kronecker-Weber Theorem]
Let $L/\Rats$ be a finite \PMlinkid{abelian extension}{AbelianExtension}, then $L$ is contained
in a cyclotomic extension, i.e. there is a root of unity $\zeta$
such that $L \subseteq \Rats(\zeta)$.
\end{thm}
In a similar fashion to this result, the theory of elliptic curves
with complex multiplication provides a classification of \PMlinkid{abelian
extensions}{AbelianExtension} of quadratic imaginary number fields:
\begin{thm}
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:
\begin{enumerate}
\item $K(j(E))$ is the Hilbert class field of $K$.
\item If $j(E)\neq 0,1728$ then the maximal \PMlinkid{abelian extension}{AbelianExtension} 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 $E$.
\end{enumerate}
\end{thm}
Note: The map $h\colon E \to \Complex$ is called a \emph{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:
$$ h(P)=
\begin{cases}
x(P) ,\text{ if $j(E)\neq 0, 1728$};\\
x^2(P) ,\text{ if $j(E)=1728$};\\
x^3(P) ,\text{ if $j(E)=0$}.
\end{cases}
$$
\begin{thebibliography}{9}
\bibitem{lang} S. Lang, {\em Algebraic Number Theory}, Springer-Verlag, New York.
\bibitem{silverman} Joseph H. Silverman, {\em Advanced Topics in the Arithmetic of Elliptic Curves}. Springer-Verlag, New
York.
\end{thebibliography}