Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 22, 2019

A note on prime divisors of polynomials P(Tk); k ≥ 1

  • François Legrand EMAIL logo
From the journal Mathematica Slovaca

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.

MSC 2010: 11R09; 11R32; 12E05; 12F10

This work is partially supported by the Israel Science Foundation (grants No. 696/13, No. 40/14, and No. 577/15).


  1. (Communicated by Milan Paštéka )

References

[1] Beckmann, S.: On extensions of number fields obtained by specializing branched coverings, J. Reine Angew. Math. 419 (1991), 27–53.10.1515/crll.1991.419.27Search in Google Scholar

[2] Dèbes, P.: Groups with no parametric Galois realizations, Ann. Sci. Éc. Norm. Supér. 51 (2018), 143–179.10.24033/asens.2353Search in Google Scholar

[3] Gerst, I.—Brillhart, J.: On the prime divisors of polynomials, Amer. Math. Monthly 78(3) (1971), 250–266.10.1080/00029890.1971.11992737Search in Google Scholar

[4] Granville, A.: Rational and integral points on quadratic twists of a given hyperelliptic curve, Int. Math. Res. Not. IMRN 8 (2007), Art. ID 027, 24 pp.10.1093/imrn/rnm027Search in Google Scholar

[5] Lang, S.: Algebra. Revised 3rd ed., Graduate Texts in Math. 211, Springer-Verlag, New York, 2002.10.1007/978-1-4613-0041-0_1Search in Google Scholar

[6] Legrand, F.: On parametric extensions over number fields, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18(2) (2018), 551–563.10.2422/2036-2145.201603_005Search in Google Scholar

[7] Legrand, F.: Specialization results and ramification conditions, Israel J. Math. 214(2) (2016), 621–650.10.1007/s11856-016-1349-ySearch in Google Scholar

[8] Legrand, F.: Twists of superelliptic curves without rational points, Int. Math. Res. Not. IMRN 2018(4) (2018), 1153–1176. 10.1093/imrn/rnw270.Search in Google Scholar

[9] Nagell, T.: Sur les diviseurs premiers des polynômes, Acta Arith. 15 (1969), 235–244.10.4064/aa-15-3-235-244Search in Google Scholar

[10] Sadek, M.: On quadratic twists of hyperelliptic curves, Rocky Mountain J. Math. 44(3) (2014), 1015–1026.10.1216/RMJ-2014-44-3-1015Search in Google Scholar

[11] Schinzel, A.: Remarque sur le travail précédent de T. Nagell, Acta Arith. 15 (1969), 245–246.10.4064/aa-15-3-245-246Search in Google Scholar

Received: 2017-08-01
Accepted: 2018-01-25
Published Online: 2019-01-22
Published in Print: 2019-02-25

© 2019 Mathematical Institute Slovak Academy of Sciences

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0215/html
Scroll to top button