# class number divisibility in cyclic extensions

In this entry, the class number of a number field $L$ is denoted by $h_{L}$.

###### Theorem 1.

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 $n$.

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.

###### Corollary 1.

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 $q$.

Note that a Galois extension $F/K$ of prime degree has no non-trivial subextensions.

Title class number divisibility in cyclic extensions ClassNumberDivisibilityInCyclicExtensions 2013-03-22 15:07:41 2013-03-22 15:07:41 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11R29 msc 11R32 msc 11R37 IdealClass ClassNumbersAndDiscriminantsTopicsOnClassGroups