The problem:
Show that $p(x)=x^2-x-1\in \mathbb Z/(3)[x]$ is irreducible over $\mathbb Z/(3)$. Show that there exists an extension K of $\mathbb Z/(3)$ with nine elements having all roots of $p(x)$.
What I did:
I almost solved the question, I proved that $p(x)$ is irreducible because there are no roots of $p(x)$ in $\mathbb Z/(3)$ I proved also the extension K is in the form: $K=\{a+bu; a,b \in \mathbb Z/(3)\}$ where $u$ is a root of $p(x)$ in an extension of $\mathbb Z/(3)$ with nine elements by Kronecker's Theorem. My problem is prove that any root of $p(x)$ is in $K$. Obviously $u \in K$ and the other root? Notice that p(x) has just two roots because $Z/(3)$ is an integral domain. Anyone can help me, please?
Thanks