Let $n>1$ be an integer. An old result of Selmer, See Theorem 1, page 289 in
http://www.mscand.dk/article.php?id=1472,
(If the link does not work try googling: selmer trinomials)
says that
$$ S(n) = x^n-x-1 $$ is irreducible over the the field $k= \mathbb{Q}$ of rational numbers.
Question : What is known about the possible irreducibility (or not) of the sligthly more general trinomial
$$ T(n,m) = x^n - x^m -1 $$
(with $0 < m < n$)
over the prime field
$$ k =GF(p) $$
such that (say)
(a) $p>2,$
(since seems there are many known results for binary polynomials)
and
(b) $n$ goes to infinity when $p$ goes to infinity.
EDIT: Observe that something can be said about the parity of the number of irreducible factors: Use Stickelberger's parity theorem.
