Splitting Field of a Polynomial over a Finite Field

Quote:

Originally Posted by slevvio

Hello everyone, I was wondering if I could get some help with this.

Find the splitting field of the polynomial $f = x^3 + 2x +1 \in \mathbb{Z}_3 [x]$

Well I know that $\mathbb{Z}_3[x] / \langle x^3 + 2x + 1 \rangle$ is a field extension containing $\alpha = x + \langle f \rangle$ which is a root of the polynomial $f$ .

But is this a splitting field ? Can there not be another element $\alpha '$ which hasn't appeared in this field extension? Thanks for any help.

Dividing $f(x)=x^3+2x+1$ by $w:=x+<f>$ , we get that

$x^3+2x+1=(x+2w)(x^2+wx+w^2+2)$ , and

since the field's characteristic is not 2 we know the above quadratic splits on $\mathbb{Z}/3\mathbb{Z}[x]/<f>$

iff its discriminant is a square. Now just chek the discriminant indeed is square in this field...

Tonio

Pd For example, $w+1$ is another root of $f(x)$ ...