Abstract:For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb {Q}^{\times }$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb {Q}$ and a union of conjugacy classes $C \subset \mathrm {Gal}(F/\mathbb {Q})$, we develop methods for determining if there exist infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra’s work. We present several applications, including a characterization of all integers $a_1, \ldots , a_n$ such that $\mathrm {ord}_p(a_1) = \ldots = \mathrm {ord}_p(a_n)$ for infinitely many primes $p$. The obtained results are conditional to a generalization of the Riemann hypothesis.

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中文翻译：

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以素数为模的一组整数的阶数相等

摘要：对于$\mathbb {Q}^{\times }$的有限生成子群$W_1, \ldots , W_t$, 整数$k_1, \ldots , k_t$, $\mathbb {Q }$ 和共轭类的并集 $C \subset \mathrm {Gal}(F/\mathbb {Q})$，我们开发了确定是否存在无限多个素数 $p$ 的方法，使得减少的指数为$W_i$ 模$p$ 除以$k_i$，$F$ 上$p$ 的Artin 符号包含在$C$ 中。结果是 HW Lenstra 工作的多变量概括。我们提出了几个应用，包括对所有整数 $a_1, \ldots , a_n$ 的表征，使得 $\mathrm {ord}_p(a_1) = \ldots = \mathrm {ord}_p(a_n)$ 对于无穷多个素数 $ p$。获得的结果以黎曼假设的推广为条件。

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