| Version 11 |
Version 10 |
| 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. |
| 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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