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.

`Finite Fields`

, vol 20 of Encyclopedia of Mathematics and its Applications, reprinted 1987) about the number $N(q,n)$ of $a \in GF(q)$ that make the trinomial $$ S_n(x) =x^n+x+a $$ irreducible over $GF(q)$: $$ \vert N(q,n)- \frac{q}{n} \vert \leq B_n q^{1/2} $$ for some constant $B_n$ depending only on $n.$ – Luis H Gallardo Mar 8 '11 at 23:46