Mathematical Research Letters

Volume 14 (2007)

Number 3

Generic Galois extensions for $\SL_2(\mathbb{F}_5)$ over $\Q$

Pages: 443 – 452

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n3.a9

Author

Bernat Plans (Universitat Politècnica de Catalunya)

Abstract

Let $G_n$ be a double cover of either the alternating group $A_n$ or the symmetric group $S_n$, and let $G_{n-1}$ be the corresponding double cover of $A_{n-1}$ or $S_{n-1}$. For every odd $n\geq 3$ and every field $k$ of characteristic $0$, we prove that the following are equivalent: {\bf (i)} there exists a generic extension for $G_{n-1}$ over $k$, {\bf (ii)} there exists a generic extension for $G_n$ over $k$. As a consequence, there exists a generic extension over $\Q$ for the group $\widetilde{A_5}\cong \SL_2(\mathbb{F}_5)$.

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