Chapter 8.6, Problem 4E

In Exercises $3-6$, a field $F$, a polynomial $p\left(x\right)$ over $F$, and an element of the field $F\left(\alpha \right)$ obtained by adjoining a zero $\alpha$ of $p\left(x\right)$ to $F$ are given. In each case:

Verify that $p\left(x\right)$ is irreducible over $F$.

Write out 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 $F\left(\alpha \right)$.

Find the multiplicative inverse of the given element of $F\left(\alpha \right)$.

$F={\mathbb{Z}}_3$, $p\left(x\right)=x^3+x^2+2x+1$, ${\alpha }^2+2\alpha +1$