Irreducible polynomials with coefficients $1$ over $GF(p)$

Question

Let $F=GF(p)$, the finite field with $p$ elements where $p$ is a prime. Is it always possible to find an irreducible polynomial of any degree with all nonzero coefficients are $1$?

Answer 1

This is not true. Take $p=13$ and consider degree $2$ polynomials which you think:

$$x^2, x^2+x, x^2+x+1, x^2+1$$ First two are not irreducible.

For third, $3$ is a root.

For fourth $5$ is a root.