Problem 5

Let $p(x)=x^{3}+2x+2\in\mathbb{Z}_{3}[x]$.

1. Show that $p(x)$ is irreducible over $\mathbb{Z}_{3}$.

2. Let $\alpha$ be a zero of $p(x)$ in the field $F=\mathbb{Z}_{3}[x]/\left\langle p(x)\right\rangle$. Show that $\alpha$ is not a generator of $F^{\ast}$.

3. Find a generator of $F^{\ast}$, and prove that your answer is indeed a generator of $F^{\ast}$. Express your answer in terms of $\alpha$.