On unramified Galois extensions of certain algebraic number fields (Q1321746)

Let \(\ell\equiv 1\bmod 8\) be a prime number, and let \(a\in \mathbb{Z}\) be a rational integer, not divisible by \(\ell\), such that the polynomial \(f(X)= x^ \ell+ aX+ a\) is irreducible over \(\mathbb{Q}\). Denote by \(\alpha_ i\) \((i=1,\dots, \ell)\) the roots in \(\mathbb{C}\) of \(f(X)=0\). Following the first author [Tokyo J. Math. 14, 227-229 (1991; Zbl 0734.11062)], if \((\ell-1)^{\ell -1} a+\ell^ \ell\) is a square in \(\mathbb{Z}\), then the Galois group of \(f(X) =0\) over \(\mathbb{Q}\) is a non-cyclic simple group, and every prime ideal is unramified in the extension \(\mathbb{Q} (\alpha_ 1, \alpha_ 2, \dots, \alpha_ \ell)/ \mathbb{Q}(\alpha_ 1)\). In this note, the authors exhibit all the integers \(a\) for \(\ell=17\).