Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II
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AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.
2019 ◽
Vol 15
(08)
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pp. 1617-1633
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1995 ◽
Vol 06
(03)
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pp. 337-370
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2014 ◽
Vol 10
(08)
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pp. 1921-1927
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1993 ◽
Vol 113
(1)
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pp. 87-90
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