This question was asked on NMBRTHRY by Kurt Foster:

If $p$ is a prime number and $\mathbb{F}_p$ the field of $p$ elements, the zeroes of the Artin-Schreier polynomial

$x^p - x - 1 \in \mathbb{F}_p[x]$

obviously have multiplicative order dividing $1 + p + p^2 + \dots + p^{p-1} = (p^p - 1)/(p-1)$ (express the norm as the product of the compositional powers of the Frobenius map)

Once upon a time, long long ago, I read that it had been conjectured (by Shafarevich IIRC) that this is the *exact* multiplicative order for every prime $p$. Can anyone supply a reference?