# Thaine’s theorem

Let $F/\mathbb{Q}$ be a totally real abelian number field. By the Kronecker-Weber theorem^{}, there exists an $m$ such that $F\subset \mathbb{Q}({\zeta}_{m})$. Let $G$ be the Galois group of the extension^{} $F/Q$. Let ${\mathcal{O}}_{F}^{\times}$ denote the group of units in the ring of integers^{} of $F$, let $C$ be the subgroup of ${\mathcal{O}}_{F}^{\times}$ consisting of units $\eta $ of the form

$\eta =\pm {N}_{\mathbb{Q}({\zeta}_{m})/F}\left({\displaystyle \prod _{a\in {(\mathbb{Z}/m\mathbb{Z})}^{\times}}}{({\zeta}_{m}^{a}-1)}^{{b}_{a}}\right)$ |

for some collection^{} of ${b}_{a}\in \mathbb{Z}$. (Here, $N$ denotes the norm operator and ${\zeta}_{m}$ is a primitive $m$-th root of unity^{}.) Finally, let $A$ denote the ideal class group^{} of $F$.

###### Theorem 1 (Thaine).

Suppose $p$ is a rational prime not dividing the degree $\mathrm{[}F\mathrm{:}\mathrm{Q}\mathrm{]}$ and suppose $\theta \mathrm{\in}\mathrm{Z}\mathit{}\mathrm{[}G\mathrm{]}$ annihilates the Sylow $p$-subgroup of $E\mathrm{/}{C}^{\mathrm{\prime}}$. Then $\mathrm{2}\mathit{}\theta $ annihilates the Sylow $p$-subgroup of $A$.

This is one of the most sophisticated results concerning the annihilators^{} of an ideal class group. It is a direct, but more complicated, version of Stickelberger’s theorem, applied to totally real fields (for which Stickelberger’s theorem gives no information).

Title | Thaine’s theorem |
---|---|

Canonical name | ThainesTheorem |

Date of creation | 2013-03-22 14:12:34 |

Last modified on | 2013-03-22 14:12:34 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 11R29 |