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See detailOn the order of the reductions of algebraic numbers
Sgobba, Pietro UL

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 ▲]

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See detailKummer theory for number fields and the reductions of algebraic numbers II
Perucca, Antonella UL; Sgobba, Pietro UL

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 ▲]

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See detailExplicit Kummer theory for the rational numbers
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailOn the distribution of the order and index for the reductions of algebraic numbers
Sgobba, Pietro UL

E-print/Working paper (2020)

Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a torsion-free subgroup of rank r in K*. We investigate under GRH the number of primes p of K such that each of the orders of ... [more ▼]

Let \alpha_1,...,\alpha_r be algebraic numbers in a number field K generating a torsion-free 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 ▲]

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See detailKummer theory for number fields
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

in Proceedings of the Roman Number Theory Association (2020)

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See detailAddendum to: Reductions of algebraic integers
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailKummer theory for number fields and the reductions of algebraic numbers
Perucca, Antonella UL; Sgobba, Pietro UL

E-print/Working paper (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 ▲]

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See detailThe degree of Kummer extensions of number fields
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 torsion-free subgroup of K* of positive rank r. Consider the degree of the cyclotomic-Kummer extension K(\zeta_n ... [more ▼]

Let K be a number field, and let \alpha_1,..., \alpha_r be elements of K* which generate a torsion-free subgroup of K* of positive rank r. Consider the degree of the cyclotomic-Kummer extension K(\zeta_n, \sqrt[n_1]{\alpha_1},..., \sqrt[n_r]{\alpha_r}), where n_i divides n for all i. We prove the following result: there is an integer x such that all above 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) for all i. Moreover, we prove that if n, n_1,..., n_r are powers of some prime number \ell, then there are formulas for the Kummer degrees over K(\zeta_n) that only depend on finitely many computable parameters. [less ▲]

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See detailKummer theory for number fields via entanglement groups
Perucca, Antonella UL; Sgobba, Pietro UL; Tronto, Sebastiano UL

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 ▲]

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See detailExplicit Kummer theory for quadratic fields
Hörmann, Fritz; Perucca, Antonella UL; Sgobba, Pietro UL 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 ▲]

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