## Abstract

Let K be a number field, and let G be a finitely generated subgroup of K^{×}. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes p of K such that the order of (G mod p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes p for which the order is k-free, and those for which the order has a prescribed l-adic valuation for finitely many primes l. An additional condition on the Frobenius conjugacy class of p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.

Original language | English |
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Pages (from-to) | 2281-2305 |

Number of pages | 25 |

Journal | Mathematics of Computation |

Volume | 92 |

Issue number | 343 |

DOIs | |

Publication status | Published - Sept 2023 |

Externally published | Yes |

## Keywords

- Chebotarev density theorem
- Reductions of algebraic numbers
- distribution of primes
- multiplicative order
- natural density