Does $\mathbb{F}_9$ contain a 4th root of unity? I realised that I don't know how to construct $\mathbb{F}_9$. I'm guessing that $\mathbb{F}_9 = \mathbb{F_3(\theta)}$, where $\theta$ is the root of some irreducible polynomial over $\mathbb{F}_3[x]$ of degree two? 
Must I even construct $\mathbb{F}_9$ in order to determine whether it contains a 4th root of unity or is there some other simpler way I'm missing?

$$\mathbb F_9\cong\mathbb F_3[X]/(X^2+1)\cong \mathbb F_3[\alpha ],$$
where $\alpha ^2=-1$. In particular, $\alpha ^4=1$.

Just for information, $$\mathbb F_9=\{0,1,2,\alpha ,2\alpha ,1+\alpha ,2+\alpha ,1+2\alpha ,2+2\alpha \}.$$
