Possible Duplicate:

Multiplicative order of zeros of the Artin-Schreier Polynomial

I will be grateful for any reference to some literature on the following question (to the best of my knowledge the answer is not known):

Is multiplicative order of roots of $x^p - x - 1$ equal to $\frac{p^p-1}{p-1}$ over $GF(p)$?

Brief Note: $P(x) = x^p - x - 1$ is irreducible. (Proof: Observe that composition with $x+1$ does not change $P$, so if polynomial Q(x) is a factor than Q(x+a) is also a factor.) Norm($x$) = 1, so the maximum possible order for $x$ is $\frac{p^p-1}{p-1}$.

a root of$x^p-x-1$... – Zev Chonoles Dec 26 '10 at 6:51