![]() Perucca, Antonella ![]() in Rivista di Matematica della Universita di Parma (in press) Detailed reference viewed: 66 (2 UL)![]() Perucca, Antonella ![]() ![]() ![]() 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: 107 (12 UL)![]() Perucca, Antonella ![]() ![]() ![]() 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: 185 (29 UL)![]() Perucca, Antonella ![]() in Convergence (2020) Detailed reference viewed: 63 (8 UL)![]() ; Perucca, Antonella ![]() in Glasgow Mathematical Journal (2020) Detailed reference viewed: 18 (3 UL)![]() Begalla, Engjell ![]() ![]() Article for general public (2020) Detailed reference viewed: 73 (0 UL)![]() Perucca, Antonella ![]() ![]() ![]() in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 61 (9 UL)![]() Perucca, Antonella ![]() in Uitwiskeling (2020) Detailed reference viewed: 44 (1 UL)![]() Perucca, Antonella ![]() Article for general public (2020) Detailed reference viewed: 80 (4 UL)![]() Perucca, Antonella ![]() in Beiträge zum Mathematikunterricht 2020 (2020) Detailed reference viewed: 37 (8 UL)![]() Perucca, Antonella ![]() ![]() 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: 132 (19 UL)![]() Perucca, Antonella ![]() ![]() ![]() 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: 126 (24 UL)![]() Perucca, Antonella ![]() Article for general public (2020) Detailed reference viewed: 29 (5 UL)![]() Perucca, Antonella ![]() Article for general public (2019) Detailed reference viewed: 46 (9 UL)![]() Perucca, Antonella ![]() ![]() 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: 231 (41 UL)![]() Perucca, Antonella ![]() in Proceedings of the Roman Number Theory Association (2019), 4 Detailed reference viewed: 102 (9 UL)![]() Perucca, Antonella ![]() Article for general public (2019) Detailed reference viewed: 6 (1 UL)![]() Perucca, Antonella ![]() in Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019) (2019) Detailed reference viewed: 88 (6 UL)![]() ; Perucca, Antonella ![]() in Journal of the Institute of Mathematics of Jussieu (2019) Detailed reference viewed: 49 (6 UL)![]() Perucca, Antonella ![]() in Rivista di Matematica della Universita di Parma (2019) Detailed reference viewed: 77 (14 UL) |
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