Chapter 8.6, Problem 3E

(a)

To determine

To prove: $p(x)=x^3+2x^2+1$ is irreducible over field $F={\mathbb{Z}}_3$.

(b)

To determine

A formula for the product of two arbitrary elements $a_0+a_1\alpha +a_2{\alpha }^2$ and $b_0+b_1\alpha +b_2{\alpha }^2$ of $F(\text{}\alpha )$.

(c)

To determine

The multiplicative inverse of the element ${\alpha }^2+\alpha +2$ of $F(\text{}\alpha )$.