Let K be a purely inseparable extension of a field k of characteristic p ≠ 0. Suppose that $[k:k^p]$ is finite. We recall that K/k is lq-modular if K is modular over a finite extension of k. Moreover, there exists a smallest extension m/k (resp. M/K) such that K/m (resp. M/k) is lq-modular. Our main result states the existence of a greatest lq-modular and relatively perfect subextension of K/k. Other results can be summarized in the following: 1. The product of lq-modular extensions over k is lq-modular...

On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.

This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is...

In this paper we investigate Hesse’s elliptic curves $H_{\mu }:U^3+V^3+W^3=3\mu UVW,\mu \in \mathbf{Q}-\left\{1\right\}$, and construct their twists, $H_{\mu ,t}$ over quadratic fields, and $\tilde{H}(\mu ,t),\mu ,t\in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that $H_{\mu }$ is a twist of $\tilde{H}(\mu ,t)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R(t;X):=X^3+tX+t,t\in \mathbf{Q}-\{0,-27/4\}$, to parametrize all of quadratic fields and cubic ones. It should be noted that $\tilde{H}(\mu ,t)$ is a twist of $H_{\mu }$ as algebraic curves because it may not always have any rational points...

A (monic) polynomial $f\left(x\right)\in \mathbb{Z}\left[x\right]$ is called intersective if the congruence $f\left(x\right)\equiv 0$ mod $m$ has a solution for all positive integers $m$. Call $f\left(x\right)$nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group $G$ can be realized as the Galois group over $\mathbb{Q}$ of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable $G$ reduces to the ordinary...