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 ${ℰ}_k$.

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.

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 ⊆ $${\overline{\mathbb{Q}}}_p$$ .

Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r(x,t)$ defined over a function field $K\left(t\right)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of generalized Rikuna polynomials${\left\{r_n(x,t)\right\}}_{n\ge 1}$ of degree ${\ell }^n$. The $r_n(x,t)$ are constructed iteratively from the $r(x,t)$. We compute the Galois groups of the $r_n(x,t)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.

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 $p$.