<s>[INST] Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial 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 $x^4+x^3+x+2$
 [/INST] Hint: Note that $x^4+x^3+x+2=(x^2+1)(x^2+x+2)$ and $(a+1)^2+(a+1)+2=0$.
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