A note on prime divisors of polynomials P(Tk); k ≥ 1
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Abstract Let F be a number field, OF the integral closure of ℤ in F, and P(T) ∈ OF[T] a monic separable polynomial such that P(0) ≠ 0 and P(1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of OF such that the reduction modulo 𝓟 of P(T) has a root in the residue field OF/𝓟, but the reduction modulo 𝓟 of P(Tk) has no root in OF/𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.
2003 ◽
Vol 2003
(71)
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pp. 4455-4464
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2021 ◽
Vol 14
(2)
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pp. 380-395
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2014 ◽
Vol 10
(04)
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pp. 885-903
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2018 ◽
Vol 2018
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pp. 1-9
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