### $\mathbf{Q}$-algebre $p$-adiche e loro rappresentazioni

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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 ${\mathcal{E}}_{k}$ 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 ${\mathcal{E}}_{{\mathbb{Q}}_{p}}$ 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 ${\mathcal{E}}_{k}$.

Let K be a nonarchimedean field, and let ϕ ∈ K(z) be a polynomial or rational function of degree at least 2. We present a necessary and sufficient condition, involving only the fixed points of ϕ and their preimages, that determines whether or not the dynamical system ϕ: ℙ¹ → ℙ¹ has potentially good reduction.

We present an algorithm that returns a proper factor of a polynomial $\Phi \left(x\right)$ over the $p$-adic integers ${\mathbb{Z}}_{p}$ (if $\Phi \left(x\right)$ is reducible over ${\mathbb{Q}}_{p}$) or returns a power basis of the ring of integers of ${\mathbb{Q}}_{p}\left[x\right]/\Phi \left(x\right){\mathbb{Q}}_{p}\left[x\right]$ (if $\Phi \left(x\right)$ is irreducible over ${\mathbb{Q}}_{p}$). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

Let $k$ be a number field. In this paper, we give a formula for the mean value of the square of class number times regulator for certain families of quadratic extensions of $k$ characterized by finitely many local conditions. We approach this by using the theory of the zeta function associated with the space of pairs of quaternion algebras. We also prove an asymptotic formula of the correlation coefficient for class number times regulator of certain families of quadratic extensions.

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