Chapter 8.6, Problem 7E

(a)

To determine

All elements of the field ${\mathbb{Z}}_3(\alpha )$ that is obtained by adjoining a zero $\alpha$ of $p(x)$ to ${\mathbb{Z}}_3$, where $p(x)=x^3+2x^2+1$ is an irreducible polynomial over ${\mathbb{Z}}_3$.

(b)

To determine

All elements of the field ${\mathbb{Z}}_3(\alpha )$ that is obtained by adjoining a zero $\alpha$ of $p(x)$ to ${\mathbb{Z}}_3$, where $p(x)=x^3+x^2+2x+1$ is an irreducible polynomial over ${\mathbb{Z}}_3$.