Irreducibility of some trinomials modulo $p$

2011-02-24T09:20:32Z

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.

Answer by Aaron Meyerowitz for Irreducibility of some trinomials modulo $p$

2011-02-25T06:41:24Z

Based on a small number of small cases I suspect that the majority of those of those do factor.

Let $r$ be a primitive root $\mod p$ then there is an $n$ such that $r^n=r+1 \mod p$. Then $x-r$ is a factor of $x^n-x-1$ in $\mathbb{Z}_p$. On average $n$ should be about $p/2$. Of course one can use $x^{n+p-1}-x-1$ but that seems like cheating. That is just cases with a linear factor.

For $p=19, $ $x^k-x-1$ factors for $2 \le k \le 18$ except for $k=4$ and $k=15$.

Irreducibility of some trinomials modulo $p$ - MathOverflow

Irreducibility of some trinomials modulo $p$

Answer by Aaron Meyerowitz for Irreducibility of some trinomials modulo $p$