Recall that $F_q[x]/(p(x))$ is a field if and only if $p$ is irreducible. However, the polynomial in question is reducible: $x^9+x^5+x^3+x+1=\left(1+x+x^2\right) \left(1+x^2+x^4+x^6+x^7\right)$, so $F_q[x] /(x^9+x^5+x^3+x+1)$ cannot be a field.