On the product of the conjugates outside the unit circle of an algebraic number
On the S-Euclidean minimum of an ideal class
We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. In particular, we resolve a conjecture of Lenstra except when the S-units have rank one. The proof is self-contained but uses ideas from...
On the splitting of primes in an arithmetic progression. II
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