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Multiplicative order and Frobenius symbol for the reductions of number fields. (English) Zbl 1436.11138

Balakrishnan, Jennifer S. (ed.) et al., Research directions in number theory. Women in numbers IV. Proceedings of the women in numbers, WIN4 workshop. Banff International Research Station, Banff, Alberta, Canada, August 14–18, 2017. Cham: Springer. Assoc. Women Math. Ser. 19, 161-171 (2019).
Summary: Let \(L/K\) be a finite Galois extension of number fields, and let \(G\) be a finitely generated subgroup of \(K^\times \). We study the natural density of the set of primes of \(K\) having some prescribed Frobenius symbol in \(\operatorname{Gal}(L/K)\), and for which the reduction of \(G\) has multiplicative order with some prescribed \(\ell \)-adic valuation for finitely many prime numbers \(\ell \). This extends in several directions results by P. Moree and B. Sury [Int. J. Number Theory 5, No. 4, 641–665 (2009; Zbl 1190.11052)] and by K. Chinen and C. Tamura [Tokyo J. Math. 35, No. 2, 441–459 (2012; Zbl 1276.11149)], and has to be compared with the very general result of V. Ziegler [Unif. Distrib. Theory 1, No. 1, 65–85 (2006; Zbl 1147.11054)].
For the entire collection see [Zbl 1428.11002].

MSC:

11R44 Distribution of prime ideals
11R20 Other abelian and metabelian extensions
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References:

[1] K. Chinen and C. Tamura, On a distribution property of the residual order of a(mod p) with a quadratic residue condition, Tokyo J. Math.35 (2012), 441-459. · Zbl 1276.11149 · doi:10.3836/tjm/1358951329
[2] C. Debry and A. Perucca, Reductions of algebraic integers, J. Number Theory, 167 (2016), 259-283. · Zbl 1411.11120 · doi:10.1016/j.jnt.2016.03.001
[3] H. Hasse, Über die Dichte der Primzahlenp, für die eine vorgegebene ganzrationale Zahla ≠ 0 von durch eine vorgegebene Primzahll ≠ 2 teilbarer bzw. unteilbarer Ordnung modpist, Math. Ann. 162 (1965/1966), 74-76. · Zbl 0135.10203 · doi:10.1007/BF01361933
[4] H. Hasse, Über die Dichte der Primzahlenp, für die eine vorgegebene ganzrationale Zahla ≠ 0 von gerader bzw. ungerader Ordnung modpist, Math. Ann. 166 (1966), 19-23. · Zbl 0139.27501 · doi:10.1007/BF01361432
[5] P. Moree, Artin’s primitive root conjecture – a survey, Integers12A (2012), No. 6, 1305-1416. · Zbl 1271.11002
[6] P. Moree and B. Sury, Primes in a prescribed arithmetic progression dividing the sequence \(\{a^k+b^k\}_{k=1}^{\infty }\), Int. J. Number Theory5 (2009), 641-665. · Zbl 1190.11052 · doi:10.1142/S1793042109002316
[7] A. Perucca, Reductions of algebraic integers II, I. I. Bouw et al. (eds.), Women in Numbers Europe II, Association for Women in Mathematics Series 11 (2018), 10-33. · Zbl 1414.11156
[8] V.
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