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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: 158 (27 UL)Kummer theory for number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano in Proceedings of the Roman Number Theory Association (2020) Detailed reference viewed: 49 (9 UL)Addendum to: Reductions of algebraic integers Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano E-print/Working paper (2019) 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: 120 (22 UL)The degree of Kummer extensions of number fields Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano E-print/Working paper (n.d.) 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. Moreover, if n, n_1, ... , n_r are powers of some prime number, then we prove that one can compute parametric formulas (with parameters n, n_1, ... , n_r) for the degree of the above extensions. [less ▲] Detailed reference viewed: 75 (8 UL)Kummer theory for number fields via entanglement groups Perucca, Antonella ; Sgobba, Pietro ; Tronto, Sebastiano E-print/Working paper (n.d.) Let $K$ be a number field, and let $G$ be a finitely generated and torsion-free subgroup of $K^\times$. We are interested in computing the degree of the cyclotomic-Kummer extension $K(\sqrt[n]{G})$ over ... [more ▼] Let $K$ be a number field, and let $G$ be a finitely generated and torsion-free subgroup of $K^\times$. We are interested in computing the degree of the cyclotomic-Kummer extension $K(\sqrt[n]{G})$ over $K$, where $\sqrt[n]{G}$ consists of all $n$-th roots of the elements of $G$. We develop the theory of entanglements introduced by Lenstra, and apply it to compute the above degrees. [less ▲] Detailed reference viewed: 83 (3 UL)Explicit Kummer theory for quadratic fields ; Perucca, Antonella ; Sgobba, Pietro et al E-print/Working paper (n.d.) Let K be a number field, and let \alpha \in K be an algebraic number which is neither 0 nor a root of unity. The ratio between n and the Kummer degree [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)], where n ... [more ▼] Let K be a number field, and let \alpha \in K be an algebraic number which is neither 0 nor a root of unity. The ratio between n and the Kummer degree [K(\zeta_m,\sqrt[n]{\alpha}):K(\zeta_m)], where n divides m, is known to be bounded independently of n and m. For some families of quadratic number fields we describe an explicit algorithm that provides formulas for all those Kummer degrees. In particular, for the given fields we are able to give explicit expressions for the Kummer extensions contained in a cyclotomic extension. [less ▲] Detailed reference viewed: 63 (6 UL)Arithmetic Billiards Perucca, Antonella ; ; Tronto, Sebastiano E-print/Working paper (n.d.) Detailed reference viewed: 33 (0 UL)Effective Kummer Theory for Elliptic Curves ; Tronto, Sebastiano E-print/Working paper (n.d.) Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform ... [more ▼] Let E be an elliptic curve defined over a number field K, let α ∈ E(K) be a point of infinite order, and let N −1 α be the set of N -division points of α in E(K). We prove strong effective and uniform results for the degrees of the Kummer extensions [K(E[N ], N −1 α) : K(E[N ])]. When K = Q, and under a minimal (necessary) assumption on α, we show that the inequality [Q(E[N ], N −1 α) : Q(E[N ])] ≥ cN 2 holds with a constant c independent of both E and α. [less ▲] Detailed reference viewed: 29 (5 UL) |
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