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 $p$.