class number divisibility in $p$-extensionsClassNumberDivisibilityInPExtensions2005-03-10 11:24:562006-06-10 13:33:52Theoremclass number divisibility in extensions% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Gal}{\operatorname{Gal}}In this entry, the class number of a number field $F$ is denoted by $h_F$.
\begin{thm}Let $p$ be a fixed prime number.
\begin{itemize}
\item Let $F/K$ be a Galois extension with Galois group $\Gal(F/K)$ and suppose $F/K$ is a $p$-extension (so $\Gal(F/K)$ is a $p$-group). Assume that there is at most one prime or archimedean place which ramifies in $F/K$. If $h_F$ is divisible by $p$ then $h_K$ is also divisible by $p$.\\
\item Let $F/\Rats$ be a Galois extension of the rational numbers and assume that $\Gal(F/\Rats)$ is a $p$-group and at most one place (finite or infinite) ramifies then $h_F$ is not divisible by $p$.
\end{itemize}
\end{thm}