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Reductions of algebraic numbers and Artin's conjecture on primitive roots Sgobba, Pietro Doctoral thesis (2022) Detailed reference viewed: 33 (4 UL)Explicit Kummer generators for cyclotomic extensions ; Perucca, Antonella ; Sgobba, Pietro et al in JP Journal of Algebra, Number Theory and Applications (2022) Detailed reference viewed: 129 (16 UL)On the distribution of the order and index for the reductions of algebraic numbers Sgobba, Pietro in Journal of Number Theory (2021) Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of (\alpha_i mod ... [more ▼] Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of (\alpha_i mod p) lies in a given arithmetic progression associated to (\alpha_i). We also study the primes p for which the index of (\alpha_i mod p) is a fixed integer or lies in a given set of integers for each i. An additional condition on the Frobenius conjugacy class of p may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case r=1 of this problem. [less ▲] Detailed reference viewed: 96 (15 UL)The degree of Kummer extensions of number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in International Journal of Number Theory (2021) Let K be a number field, and let \alpha_1, ... , \alpha_r be elements of K* which generate a subgroup of K* of rank r. Consider the cyclotomic-Kummer extensions of K given by K(\zeta_n, \sqrt[n_1]{\alpha ... [more ▼] Let K be a number field, and let \alpha_1, ... , \alpha_r be elements of K* which generate a subgroup of K* of rank r. Consider the cyclotomic-Kummer extensions of K given by K(\zeta_n, \sqrt[n_1]{\alpha_1}, ... , \sqrt[n_r]{\alpha_r}), where n_i divides n for all i. There is an integer x such that these extensions have maximal degree over K(\zeta_g, \sqrt[g_1]{\alpha_1}, ... , \sqrt[g_r]{\alpha_r}), where g=\gcd(n,x) and g_i=\gcd(n_i,x). We prove that the constant x is computable. This result reduces to finitely many cases the computation of the degrees of the extensions K(\zeta_n, \sqrt[n_1]{\alpha_1}, ... , \sqrt[n_r]{\alpha_r}) over K. [less ▲] Detailed reference viewed: 202 (12 UL)Kummer theory for number fields via entanglement groups Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Manuscripta Mathematica (2021) Detailed reference viewed: 153 (6 UL)Explicit Kummer theory for quadratic fields ; Perucca, Antonella ; Sgobba, Pietro et al in JP Journal of Algebra, Number Theory and Applications (2021) Let K be a quadratic number field and let \alpha \in K*. We present an explicit finite procedure to compute at once all Kummer degrees [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)] for n,m \geq 1 with n|m ... [more ▼] Let K be a quadratic number field and let \alpha \in K*. We present an explicit finite procedure to compute at once all Kummer degrees [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)] for n,m \geq 1 with n|m, where \zeta_m denotes a primitive m-th root of unity. We can also replace \alpha by any finitely generated subgroup of K*. [less ▲] Detailed reference viewed: 265 (13 UL)On the order of the reductions of algebraic numbers Sgobba, Pietro Presentation (2020, February 06) Let 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 ... [more ▼] Let 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. The density of primes for which this condition holds exists (this generalizes a result of Ziegler from 2006) and it is, under certain assumptions, a computable positive rational number. We also present a uniformity property concerning some special cases. This is a joint work with A. Perucca. [less ▲] Detailed reference viewed: 63 (11 UL)Kummer theory for number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 88 (10 UL)Addendum to: Reductions of algebraic integers Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Journal of Number Theory (2020) Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. We consider Kummer extensions of G of the form K(\zeta_{2^m}, \sqrt[2^n]G)/K(\zeta_{2^m}), where n \leq m. In ... [more ▼] Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K*. We consider Kummer extensions of G of the form K(\zeta_{2^m}, \sqrt[2^n]G)/K(\zeta_{2^m}), where n \leq m. In the paper "Reductions of algebraic integers" (J. Number Theory, 2016) by Debry and Perucca, the degrees of those extensions have been evaluated in terms of divisibility parameters over K(\zeta_4). We prove how properties of G over K explicitly determine the divisibility parameters over K(\zeta_4). This result has a clear computational advantage, since no field extension is required. [less ▲] Detailed reference viewed: 160 (25 UL)Kummer theory for number fields and the reductions of algebraic numbers II Perucca, Antonella ; Sgobba, Pietro in Uniform Distribution Theory (2020) Let 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 ... [more ▼] Let 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 \ell^e is a prime power and a is a multiple of \ell (and a is a multiple of 4 if \ell=2), then the density of primes p of K such that the order of (G mod p) is congruent to a modulo \ell^e only depends on a through its \ell-adic valuation. [less ▲] Detailed reference viewed: 169 (19 UL)Explicit Kummer theory for the rational numbers Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in International Journal of Number Theory (2020) Let G be a finitely generated multiplicative subgroup of Q* having rank r. The ratio between n^r and the Kummer degree [Q(\zeta_m,\sqrt[n]{G}) : Q(\zeta_m)], where n divides m, is bounded independently of ... [more ▼] Let G be a finitely generated multiplicative subgroup of Q* having rank r. The ratio between n^r and the Kummer degree [Q(\zeta_m,\sqrt[n]{G}) : Q(\zeta_m)], where n divides m, is bounded independently of n and m. We prove that there exist integers m_0, n_0 such that the above ratio depends only on G, \gcd(m,m_0), and \gcd(n,n_0). Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction). [less ▲] Detailed reference viewed: 242 (31 UL)Kummer theory for number fields and the reductions of algebraic numbers Perucca, Antonella ; Sgobba, Pietro in International Journal of Number Theory (2019) For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov-Ribet method) of the fact that if G is ... [more ▼] For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov-Ribet method) of the fact that if G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r, then the ratio between n^r and the Kummer degree [K(\zeta_n,\sqrt[n]{G}):K(\zeta_n)] is bounded independently of n. We then apply this result to generalise to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered). [less ▲] Detailed reference viewed: 279 (42 UL)Prime divisors of the l-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level l ; Sgobba, Pietro E-print/Working paper (n.d.) Let \ell be any fixed prime number. We define the \ell-Genocchi numbers by G_n:=\ell(1-\ell^n)B_n, with B_n the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's ... [more ▼] Let \ell be any fixed prime number. We define the \ell-Genocchi numbers by G_n:=\ell(1-\ell^n)B_n, with B_n the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is \ell-Genocchi irregular if it divides at least one of the \ell-Genocchi numbers G_2,G_4,..., G_{p-3}, and \ell-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of \ell-Genocchi irregular primes in a prescribed arithmetic progression in case \ell is odd. The case \ell=2 was already dealt with by Hu, Kim, Moree and Sha (2019). Using similar methods we study the prime factors of (1-\ell^n)B_{2n}/2n and (1+\ell^n)B_{2n}/2n. This allows us to estimate the number of primes p\leq x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level \ell. [less ▲] Detailed reference viewed: 54 (3 UL)Divisibility conditions on the order of the reductions of algebraic numbers Sgobba, Pietro E-print/Working paper (n.d.) Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes \p of K such that the order of (G mod \p) is ... [more ▼] Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes \p of K such that the order of (G mod \p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes \p for which the order is k-free, and those for which the order has a prescribed \ell-adic valuation for finitely many primes \ell. An additional condition on the Frobenius conjugacy class of \p may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields. [less ▲] Detailed reference viewed: 99 (10 UL) |
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