push-down theorem on class numbersPushDownTheoremOnClassNumbers2005-02-23 15:26:142005-02-23 15:28:53Theoremclass number divisibility in extensions% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}As in the \PMlinkname{parent}{ClassNumberDivisibilityInExtensions} entry, given a number field $K$, the class number of $K$ is denoted by $h_K$.
\begin{thm}[Pushing-Down Theorem]
Let $E/F$ be a $p$-extension of number fields and suppose that only one prime ideal of $F$ is ramified in $E$ and that this prime is totally ramified. Then $p|h_E$ implies $p|h_F$.
\end{thm}
\begin{thebibliography}{HGM02} % '2nd argument contains the widest acronym'
\bibitem[Fr\"oh]{hgm}
A. Fr\"ohlich, \emph{On a method for the determination of class number factors in number fields}, Mathematika, 4 (1957), 113-121.
\bibitem[Iwas]{green}
K. Iwasawa, \emph{A note on Class Numbers of Algebraic Number Fields}, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258.
\end{thebibliography}