![]() ; Perucca, Antonella ![]() ![]() 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: 287 (13 UL)![]() Perucca, Antonella ![]() Article for general public (2021) Detailed reference viewed: 76 (7 UL)![]() Perucca, Antonella ![]() Article for general public (2021) Detailed reference viewed: 78 (1 UL)![]() Perucca, Antonella ![]() ![]() ![]() in Manuscripta Mathematica (2021) Detailed reference viewed: 163 (7 UL)![]() Perucca, Antonella ![]() in Rivista di Matematica della Universita di Parma (2020), 11 Detailed reference viewed: 95 (2 UL)![]() Begalla, Engjell ![]() ![]() Article for general public (2020) Detailed reference viewed: 147 (1 UL)![]() Perucca, Antonella ![]() Article for general public (2020) Detailed reference viewed: 109 (8 UL)![]() Perucca, Antonella ![]() Article for general public (2020) Detailed reference viewed: 59 (5 UL)![]() Perucca, Antonella ![]() Article for general public (2020) Detailed reference viewed: 74 (1 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: 179 (19 UL)![]() ; Perucca, Antonella ![]() in Glasgow Mathematical Journal (2020) Detailed reference viewed: 43 (3 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: 252 (31 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: 171 (25 UL)![]() Perucca, Antonella ![]() in Convergence (2020) Detailed reference viewed: 109 (8 UL)![]() Perucca, Antonella ![]() ![]() ![]() in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 94 (10 UL)![]() Perucca, Antonella ![]() in Beiträge zum Mathematikunterricht 2020 (2020) Detailed reference viewed: 61 (8 UL)![]() ; Perucca, Antonella ![]() in Journal of the Institute of Mathematics of Jussieu (2019) Detailed reference viewed: 96 (6 UL)![]() Perucca, Antonella ![]() in Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019) (2019) Detailed reference viewed: 122 (6 UL)![]() Perucca, Antonella ![]() in Proceedings of the Roman Number Theory Association (2019), 4 Detailed reference viewed: 122 (9 UL)![]() Perucca, Antonella ![]() ![]() Article for general public (2019) Detailed reference viewed: 70 (8 UL) |
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