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 $F\mathrm{/}K$ 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 $K\mathrm{\subseteq }E\mathrm{⊊}F$. 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 $p^f\equiv 1modn$ and $p^m$ is not congruent to $1$ modulo $n$ for any .

Corollary 1.

Let $F\mathrm{/}K$ be a Galois extension such that $\mathrm{[}F\mathrm{:}K\mathrm{]}\mathrm{=}q$ 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.