| Version 10 |
Version 9 |
| Let $K$ be a number field. There exists a finite extension $E$ of $K$ with the following properties: |
Let $K$ be a number field. There exists a finite extension $E$ of $K$ with the following properties: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $[E:K]=h_K$, where $h_K$ is the class number of $K$. |
\item $[E:K]=h_K$, where $h_K$ is the class number of $K$. |
| \item $E$ is Galois over $K$. |
\item $E$ is Galois over $K$. |
| \item The ideal class group of $K$ is isomorphic to the Galois group of |
\item The ideal class group of $K$ is isomorphic to the Galois group of |
| $E$ over $K$. |
$E$ over $K$. |
| \item Every ideal of $\rai{K}$ is a principal ideal of the ring extension $\rai{E}$. |
\item Every ideal of $\rai{K}$ is a principal ideal of the ring extension $\rai{E}$. |
| \item Every prime ideal ${\cal P}$ of $\rai{K}$ decomposes into the product of |
\item Every prime ideal ${\cal P}$ of $\rai{K}$ decomposes into the product of |
| $\frac{h_K}{f}$ prime ideals in $\rai{E}$, where $f$ is the order |
$\frac{h_K}{f}$ prime ideals in $\rai{E}$, where $f$ is the order |
| of $[{\cal P}]$ in the ideal class group of $\rai{E}$. |
of $[{\cal P}]$ in the ideal class group of $\rai{E}$. |
| \end{enumerate} |
\end{enumerate} |
| There is a unique field $E$ satisfying the above five properties, and it is known as the {\em Hilbert class field} of $K$. |
There is a unique field $E$ satisfying the above five properties, and it is known as the {\em Hilbert class field} of $K$. |
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The field $E$ may also be characterized as the maximal abelian unramified extension of $K$. Note that in this context, 'unramified' is meant not only for the finite places (the classical ideal theoretic {\pmlinkescapetext interpretation}) but also for the infinite places. That is, every real embedding of $K$ extends to a real embedding of $E$. As an example of why this is necessary, consider some real quadratic field.
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The field $E$ may also be characterized as the maximal abelian unramified extension of $K$. Note that in this context, 'unramified' is meant not only for the finite places (the classical ideal theoretic interpretation) but also for the infinite places. That is, every real embedding of $K$ extends to a real embedding of $E$. As an example of why this is necessary, consider some real quadratic field.
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