For which $\mathbb{Z}$-irreducible polynomials $\chi$ is there a nonconstant polynomial $p \equiv \pm1$ (mod $\chi$) with coefficients in $\{-1,0,1\}$?
As an example, if $\chi = x^2 + 2x + 2$, then the following polynomials are nonconstant, have coefficients in $\{-1,0,1\}$ and are congruent to $\pm1$
$x^3 + x^2 - 1 \equiv 1$
$x^4 + x^3 + x^2 + 1 \equiv -1$
Thanks in advance!
(Also, if it's helpful, all the $\chi$ I'm interested in are of the form $x^n + 2g$ for some other polynomial $g$ with constant term $\pm1$, just like the above $\chi$)