Non-parametricity of rational translates of regular Galois extensions

Joachim König¹

¹ Technion I.I.T. 32000 Haifa, Israel

Acta Arithmetica, Tome 179 (2017) no. 3, pp. 267-275

Résumé

We generalize a result of F. Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field of $g(X)-t$ over $K(t)$ is non-$G$-parametric, i.e. not all Galois extensions of $K$ with group $G$ arise as specializations of $F(s)|K(s)$.

Détail

DOI : 10.4064/aa170112-15-5

Keywords: generalize result nbsp legrand about existence non parametric galois extensions given group precisely k regular galois extension consider translates extension rational function fields other words root t rational function k regular galois extension group number field nbsp degree almost density sense rational functions degree nbsp translate root field t non g parametric galois extensions group arise specializations