Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial

Question

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_3[x]$.

We symbolize as $a$ a root of $f(x)$ in an algebraic closure $\overline{\mathbb{Z}}_3$ of $\mathbb{Z}_3$.

Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial $x^4+x^3+x+2 \in \mathbb{Z}_3[x]$.

I have done the following:

$a$ is a root of $f(x): a^2-1=0\Rightarrow a^2=1$

$x^4+x^3+x+2=x^2x^2+xx^2+x+2$

Setting $x=a$ we hav3 that:

$a^2a^2+aa^2+a+2=1+a+a+2=2a+3$

Does this mean that $a$ ios not a root of $x^4+x^3+x+2$

Answer 1

Hint: Note that $x^4+x^3+x+2=(x^2+1)(x^2+x+2)$ and $(a+1)^2+(a+1)+2=0$.