On Unramified Galois Extensions of Certain Algebraic Number Fields

Abstract

Let $a\in\mathbf{Z}$ such that $a\neq 1$, $a\neq-2^{17}$ and $(17,a)=1$. Let $\alpha_1,\alpha_2,\ldots,\alpha_{17}$ denote the roots of $x^{17}+ax+a=0$. It is shown that every prime ideal is unramified in $\mathbf{Q}(\alpha_1,\alpha_2,\ldots,\alpha_{17})/\mathbf{Q}(\alpha_1)$ if and only if $a=2^{62}n^2+4605612312119580521n+1149886651258880054$ for some $n\in\mathbf{Z}$.