class number divisibility in cyclic extensionsClassNumberDivisibilityInCyclicExtensions2005-03-10 11:43:092005-03-10 11:43:09Theoremclass number divisibility in extensions% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}In this entry, the class number of a number field $L$ is denoted by $h_L$.
\begin{thm}
Let $F/K$ be a cyclic Galois extension of degree $n$. Let $p$ be a prime such that $n$ is not divisible by $p$, and assume that $p$ does not divide $h_E$, the class number of any intermediate field $K\subseteq E \subsetneq F$. If $p$ divides $h_F$ then $p^f$ also divides $h_F$, where $f$ is the multiplicative order of $p$ modulo $n$.
\end{thm}
Recall that the multiplicative order of $p$ modulo $n$ is a number $f$ such that $p^f\equiv 1 \mod n$ and $p^m$ is not congruent to $1$ modulo $n$ for any $1\leq m <f$.
\begin{cor}
Let $F/K$ be a Galois extension such that $[F:K]=q$ is a prime distinct from the prime $p$. Assume that $p$ does not divide $h_K$. If $p$ divides $h_F$ then $p^f$ divides $h_F$, where $f$ is the multiplicative order of $p$ modulo $q$.
\end{cor}
Note that a Galois extension $F/K$ of prime degree has no non-trivial subextensions.