Kummer theory for number fields via entanglement groups

E-print/Working paper (in press)

Explicit Kummer theory for quadratic fields

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 ▲]

On the order of the reductions of algebraic numbers

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 ▲]

Addendum to: Reductions of algebraic integers

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 ▲]

Explicit Kummer theory for the rational numbers

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 ▲]

Divisibility conditions on the order of the reductions of algebraic numbers

E-print/Working paper (n.d.)

Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes p of K such that the order ord_p(G) of (G mod p ... [more ▼]

Let K be a number field, and let G be a finitely generated subgroup of K*. Without relying on (GRH) we prove an asymptotic formula for the number of primes p of K such that the order ord_p(G) of (G mod p) is divisible by a fixed integer. We also provide a computable and rational expression for the natural density of this set. Then we study the number of primes p for which ord_p(G) is k-free. An additional condition on the Frobenius conjugacy class of p may be considered. These results are generalizations of Pappalardi’s work concerning the case K = Q. [less ▲]

Explicit Kummer generators for cyclotomic extensions

E-print/Working paper (n.d.)