On considère un problème de plongement de corps de nombres algébriques, dont le noyau est abélien, et on suppose que les problèmes locaux correspondants sont résolubles. On montre que les conditions complémentaires de résolubilité, dites globales, sont fournies pour un nombre fini de représentations du noyau dans le groupe de classes d’idèles. Dans le cas d’un noyau cyclique, une seule suffit, et on la calcule.

Let $k$ be the algebraic closure of ${𝔽}_p$ and $K$ be the local field of formal power series with coefficients in $k$. The aim of this paper is the description of the set ${𝒴}_n$ of conjugacy classes of series of order $p^n$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means...

Let $K$ be a $𝔭$-adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L/K$ of degree $p$ to the exponents of generators of the norm group $N_{L/K}\left(L^{*}\right)$. This is applied in an algorithm for the construction of class fields of degree $p^m$, which yields an algorithm for the computation of class fields in general.