The Galois group of the Eisenstein polynomial X ⁵ + aX + a

Abstract.

Let p be a rational prime and let a be an integer which is divisible by p exactly to the first power. Then the Galois group $G = \hbox{Gal}_{{\mathbb{Q}}}(f)$ of the Eisenstein polynomial f = X ^p + aX + a is known to be either the full symmetric group S ^p or the affine group A(1, p), and it is conjectured that always G = S _p . In this note we settle this conjecture for p = 5 and, answering a question by J.-P. Serre, we show that this does not carry over when replacing the integer a by some rational number with 5-adic valuation equal to 1.