# 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 $K\mathrm{\subseteq}E\mathrm{\u228a}F$. 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.

Title | class number divisibility in cyclic extensions |
---|---|

Canonical name | ClassNumberDivisibilityInCyclicExtensions |

Date of creation | 2013-03-22 15:07:41 |

Last modified on | 2013-03-22 15:07:41 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11R29 |

Classification | msc 11R32 |

Classification | msc 11R37 |

Related topic | IdealClass |

Related topic | ClassNumbersAndDiscriminantsTopicsOnClassGroups |