# Thread: Splitting Field of a Polynomial over a Finite Field

1. ## Splitting Field of a Polynomial over a Finite Field

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

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

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

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

2. Originally Posted by slevvio
Hello everyone, I was wondering if I could get some help with this.

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

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

But is this a splitting field ? Can there not be another element $\displaystyle \alpha '$ which hasn't appeared in this field extension? Thanks for any help.
Dividing $\displaystyle f(x)=x^3+2x+1$ by $\displaystyle w:=x+<f>$ , we get that

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

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

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

Tonio

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