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.