Nombre de -classes invariantes. Application aux classes des corps abéliens
Nombre d'extensions abéliennes sur Q
The aim of this paper is to give the numbers of abelian number fields with given degree and ramification indices. We describe, also, an algorithm to compute all these fields.
Non-existence and splitting theorems for normal integral bases
We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower forces the tower to be split in a very strong sense.
Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics
Explicit normal integral bases are given for some cyclic quintic fields defined by Emma Lehmer’s parametric family of quintics.
Note on the congruence of Ankeny-Artin-Chowla type modulo p²
The results of [2] on the congruence of Ankeny-Artin-Chowla type modulo p² for real subfields of of a prime degree l is simplified. This is done on the basis of a congruence for the Gauss period (Theorem 1). The results are applied for the quadratic field ℚ(√p), p ≡ 5 (mod 8) (Corollary 1).
Note on the Hilbert 2-class field tower
Let be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).