## Abstract

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov-Ribet method) of the fact that if G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r, then the ratio between nr and the Kummer degree [K(ζn,Gn): K(ζn)] is bounded independently of n. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

Original language | English |
---|---|

Pages (from-to) | 1617-1633 |

Number of pages | 17 |

Journal | International Journal of Number Theory |

Volume | 15 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Sept 2019 |

Externally published | Yes |

## Keywords

- Kummer theory
- density
- multiplicative order
- number field
- reduction