Q: Prove that for any finite field $\mathbb{F_q}$, the ring $\mathbb{F_q}[x]/(x^9+x^5+x^3+x+1)$ cannot be a field.
Upon first glance I am really not sure where to start. Intuitively, it seems that I should be finding a way to show that $x^9+x^5+x^3+x+1$ is reducible over every finite field.
I have that: $x^9+x^5+x^3+x+1 = (x^2-x+1)(x^7+x^6-x^4+x^2+2x+1)$ but I don't see where to go from here.
Am I even on the right track? Is there some well-known theorem I've blanked on that would help me out? Thanks in advance!