The distribution of the multiplicative index of algebraic numbers over residue classes

Pieter Moree, Antonella Perucca, Pietro Sgobba
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引用次数: 0

Abstract

Let K be a number field and G a finitely generated torsion-free subgroup of \(K^\times \). Given a prime \(\mathfrak {p}\) of K we denote by \({{\,\textrm{ind}\,}}_\mathfrak {p}(G)\) the index of the subgroup \((G\bmod \mathfrak {p})\) of the multiplicative group of the residue field at \(\mathfrak {p}\). Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.

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代数数的乘法指数在残差类上的分布
让 K 是一个数域,G 是 \(K^\times \)的一个有限生成的无扭子群。给定 K 的一个素数 \(\mathfrak {p}\),我们用 \({{\textrm{ind}\,}}_\mathfrak {p}(G)\)表示在 \(\mathfrak {p}\)处的残差域乘法群的子群 \((G\bmod \mathfrak {p})\)的索引。在广义黎曼假说下,我们确定了K的素数的自然密度,对于这些素数来说,这个指数在一个规定的集合S中,并且在K的有限伽罗瓦扩展F中具有规定的弗罗贝尼斯(Frobenius)。
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.
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