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.

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 $\mathbb{Q}\subset K\subset L$ forces the tower to be split in a very strong sense.

Let $p=1+2^{e+1}q$ be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order $2^{e+1}$ (resp. order $d_{\phi }$ dividing q), and let ψₙ be an even character of conductor $p^{n+1}$ and order pⁿ. We put χ = δφψₙ, whose value is contained in $Kₙ=\mathbb{Q}\left({\zeta }_{(p-1)pⁿ}\right)$. It is well known that the Bernoulli number $B_{1,\chi }$ is not zero, which is shown in an analytic way. In the extreme cases $d_{\phi }=1$ and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: $T{r}_{n/1}\left(\xi B_{1,\chi }\right)\ne 0$ for any pⁿth root ξ...

On démontre, à partir de résultats de H.J. Godwin, H. Brunotte et F. Halter-Koch, le théorème suivant : soit $K$ un corps cubique cyclique de conducteur $m$ dont le groupe de Galois $G$ est engendré par $\sigma$; soit $E$ le groupe des unités de norme 1.Soit $ϵ\in E$, $ϵ\ne 1$, telle que $𝒮\left(ϵ\right)=\frac{1}{2}\left[\right(ϵ-{ϵ}^{\sigma }{)}^2+({ϵ}^{\sigma }-{ϵ}^{{\sigma }^2}{)}^2+({ϵ}^{{\sigma }^2}-ϵ{)}^2]$ soit minimum. Alors $ϵ$ est un $\mathbf{Z}\left[G\right]$-générateur de $E$.

The results of [2] on the congruence of Ankeny-Artin-Chowla type modulo p² for real subfields of $\mathbb{Q}\left({\zeta }_p\right)$ 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).

In this note we will determine the associated order of relative extensions of algebraic number fields, which are cyclic of prime order p, assuming that the ground field is linearly disjoint to the pth cyclotomic field, ${\mathbb{Q}}^{\left(p\right)}$. For quadratic extensions we will furthermore characterize when the ring of integers of the extension field is free over the associated order. All our proofs are quite elementary. As an application, we will determine the Galois module structure of ${\mathbb{Q}}^{\left(n\right)}/{\mathbb{Q}}^{{\left(n\right)}^{+}}$.