Let $p\ne 2$ be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over ${\mathbb{Q}}_p$, and extend it to when the base fields $K$ is an unramified extension of ${\mathbb{Q}}_p$.When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We derive a complete classification...

Let $k$ be a finite extension of ${\mathbb{Q}}_p$ and ${ℰ}_k$ be the set of the extensions of degree $p^2$ over $k$ whose normal closure is a $p$-extension. For a fixed discriminant, we show how many extensions there are in ${ℰ}_{{\mathbb{Q}}_p}$ with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalized to get a classification of the extensions in ${ℰ}_k$.

I extend the Hasse–Arf theorem from residually separable extensions of complete discrete valuation rings to monogenic extensions.

Let $S/R$ be a finite extension of discrete valuation rings of characteristic $p\>0$, and suppose that the corresponding extension $L/K$ of fields of fractions is separable and is $H$-Galois for some $K$-Hopf algebra $H$. Let ${𝔻}_{S/R}$ be the different of $S/R$. We show that if $S/R$ is totally ramified and its degree $n$ is a power of $p$, then any element $\rho$ of $L$ with $v_L\left(\rho \right)\equiv -v_L\left({𝔻}_{S/R}\right)-1(modn)$ generates $L$ as an $H$-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about $2$-primary units. We also prove a similar statement about the absolute norms of $p$-primary units, for all primes $p$.

Let p be a prime number, ℚp the field of p-adic numbers, and $${\overline{\mathbb{Q}}}_p$$ a fixed algebraic closure of ℚp. We provide an analytic version of the normal basis theorem which holds for normal extensions of intermediate fields ℚp ⊆ K ⊆ L ⊆ $${\overline{\mathbb{Q}}}_p$$ .

Nous donnons une preuve que tout automorphisme sauvagement ramifié d’un corps de séries formelles à une variable et à coefficients dans un corps parfait de caractéristique $p$ provient de la construction du corps des normes d’une ${\mathbf{Z}}_p$-extension totalement ramifiée d’un corps local de caractéristique $0$ ou $p$.

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.

Soit $N$ un entier $\ge 1$. Pour un nombre premier $p$ on note ${\mathbf{Q}}_p^{\mathrm{nr}}$ l’extension maximale non ramifiée de ${\mathbf{Q}}_p$. Supposons que $p^v$ divise exactement $N$. Alors, en utilisant les travaux de Carayol et la théorie du corps de classes local, on détermine une extension $E_v$ de ${\mathbf{Q}}_p^{\mathrm{nr}}$ sur laquelle la jacobienne $J_0$ de la courbe modulaire de $X_0\left(N\right)$ admet une réduction semi-stable, puis on donne une estimation de son degré.