Multiplicative order and Frobenius symbol for the reductions of number fields

in Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019) (2019)

Kummer theory for number fields and the reductions of algebraic numbers

in International Journal of Number Theory (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 ▲]

The PytEuk puzzle

Diverse speeches and writings (2018)

We present an original mathematical exhibit.

Arithmetic billiards

Article for general public (2018)

Math around the Clock

Article for general public (2018)

This is an article for the general public about mathematical clocks. Several original mathematical clocks are also presented.

Reductions of one-dimensional tori II

in Association for Women in Mathematics Series (2018)

Reductions of one-dimensional tori

in International Journal of Number Theory (2017)

Reductions of algebraic integers

in Journal of Number Theory (2016)

The prime divisors of the number of points on abelian varieties

in Journal de Theorie des Nombres de Bordeaux (2015)

The first-digit law

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