On the trace map between absolutely abelian number fields of equal conductor
On the verification of Clark's example of a euclidean but not norm-euclidean number field.
On units of some fields of the form
Let and be two prime integers and let be a positive odd square-free integer. Assuming that the fundamental unit of has a negative norm, we investigate the unit group of the fields .
On values of the Mahler measure in a quadratic field (solution of a problem of Dixon and Dubickas)
On Waring's problem in algebraic number fields
Optimality of the Width- Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
We consider digit expansions with an endomorphism of an Abelian group. In such a numeral system, the -NAF condition (each block of consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight admits an optimal -NAF).This result is then applied to imaginary quadratic bases, which are used for scalar multiplication in elliptic...
Ordre et indice modulo les puissance d'un idéal premier
Paramétrisation de structures algébriques et densité de discriminants
La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant , sous l’action de par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant . Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers,...
Perfect Gaussian integers
Period of a linear recurrence
Periodic Gaussian moats.
Plongements -adiques et -nombres de Weil
Nous introduisons la notion de nombre de Weil -adique par analogie avec la notion classique de nombre de Weil à l’infini ; et nous en étudions quelques propriétés en liaison avec les plongements et les valeurs absolues réelles ou -adiques des corps de nombres. En appendice, nous en tirons diverses applications à la théorie d’Iwasawa des tours cyclotomiques.
Pólya fields and Pólya numbers
A number field , with ring of integers , is said to be a Pólya field if the -algebra formed by the integer-valued polynomials on admits a regular basis. In a first part, we focus on fields with degree less than six which are Pólya fields. It is known that a field is a Pólya field if certain characteristic ideals are principal. Analogously to the classical embedding problem, we consider the embedding of in a Pólya field. We give a positive answer to this embedding problem by showing that...
Pólya fields, Pólya groups and Pólya extensions: a question of capitulation
A number field , with ring of integers , is said to be a Pólya field when the -algebra formed by the integer-valued polynomials on admits a regular basis. It is known that such fields are characterized by the fact that some characteristic ideals are principal. Analogously to the classical embedding problem in a number field with class number one, when is not a Pólya field, we are interested in the embedding of in a Pólya field. We study here two notions which can be considered as measures...
Potenzfreie Werte von Polynomen in algebraischen Zahlkörpern.
Power integral bases in cubic relative extensions.
Power integral bases in orders of composite fields.
Power integral bases in the family of simplest quartic fields.
Primes which remain irreducible in a normal field
Primitive divisors of the expression An - Bn in algebraic number fields.