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Non-parametricity of rational translates of regular Galois extensions

Volume 179 / 2017

Joachim König Acta Arithmetica 179 (2017), 267-275 MSC: Primary 11R32; Secondary 12E05, 12E25. DOI: 10.4064/aa170112-15-5 Published online: 7 July 2017

Abstract

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 $F(s)|K(s)$.

Authors

  • Joachim KönigTechnion I.I.T.
    32000 Haifa, Israel
    e-mail

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