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Showing results 21 to 40 of 88
   Explicit Kummer theory for quadratic fields; Perucca, Antonella  ; Sgobba, Pietro et alin 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)Every number is the beginning of a power of 2Perucca, Antonella Article for general public (2021) Detailed reference viewed: 76 (7 UL)Staircase numbersPerucca, Antonella Article for general public (2021) Detailed reference viewed: 78 (1 UL) Kummer theory for number fields via entanglement groupsPerucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Manuscripta Mathematica (2021) Detailed reference viewed: 163 (7 UL)The degree of non-Galois Kummer extensions of number fieldsPerucca, Antonella in Rivista di Matematica della Universita di Parma (2020), 11 Detailed reference viewed: 95 (2 UL)The ABCD of cyclic quadrilateralsBegalla, Engjell ; Perucca, Antonella Article for general public (2020) Detailed reference viewed: 147 (1 UL)The 15 puzzlePerucca, Antonella Article for general public (2020) Detailed reference viewed: 109 (8 UL)The seven bridges of KönigsbergPerucca, Antonella Article for general public (2020) Detailed reference viewed: 59 (5 UL)Four points, two distancesPerucca, Antonella Article for general public (2020) Detailed reference viewed: 74 (1 UL)Kummer theory for number fields and the reductions of algebraic numbers IIPerucca, 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: 179 (19 UL)Reductions of points on algebraic groups II; Perucca, Antonella in Glasgow Mathematical Journal (2020) Detailed reference viewed: 43 (3 UL)Explicit Kummer theory for the rational numbersPerucca, 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: 252 (31 UL)Addendum to: Reductions of algebraic integersPerucca, 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: 171 (25 UL) Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom ExercisePerucca, Antonella ; in Convergence (2020) Detailed reference viewed: 109 (8 UL)Kummer theory for number fieldsPerucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 94 (10 UL)Visualisierungen des InduktionsprinzipsPerucca, Antonella ; in Beiträge zum Mathematikunterricht 2020 (2020) Detailed reference viewed: 61 (8 UL)Reductions of points on algebraic groups; Perucca, Antonella in Journal of the Institute of Mathematics of Jussieu (2019) Detailed reference viewed: 96 (6 UL)Multiplicative order and Frobenius symbol for the reductions of number fieldsPerucca, Antonella in Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019) (2019) Detailed reference viewed: 122 (6 UL)Reductions of elliptic curvesPerucca, Antonella in Proceedings of the Roman Number Theory Association (2019), 4 Detailed reference viewed: 122 (9 UL)Trigonometry for the multiples of 3 degreesPerucca, Antonella ; Stranen, Deborah Article for general public (2019) Detailed reference viewed: 70 (8 UL) |
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