Chapter 8.6, Problem 2E

(a)

To determine

Addition and multiplication tables for this field. $p(x)=x^2+x+1$ over $F={\mathbb{Z}}_2$, and decide whether the ring ${\mathbb{Z}}_2\left[x\right]/\left(p(x)\right)$ is a field or not.

(b)

To determine

Addition and multiplication tables for this field. $p(x)=x^3+1$ over $F={\mathbb{Z}}_2$, and decide whether the ring ${\mathbb{Z}}_2\left[x\right]/\left(p(x)\right)$ is a field or not.

(c)

To determine

Addition and multiplication tables for the field $p(x)=x^3+x+1$ over $F={\mathbb{Z}}_2$, and decide whether the ring ${\mathbb{Z}}_2\left[x\right]/\left(p(x)\right)$ is a field or not.

(d)

To determine

Addition and multiplication tables for this field. $p(x)=x^3+x^2+1$ over $F={\mathbb{Z}}_2$, and decide whether the ring ${\mathbb{Z}}_2\left[x\right]/\left(p(x)\right)$ is a field or not.

(e)

To determine

Addition and multiplication tables for this field. $p(x)=x^2+x+1$ over $F={\mathbb{Z}}_2$., and decide whether the ring ${\mathbb{Z}}_2\left[x\right]/\left(p(x)\right)$ is a field or not.

(f)

To determine

All the elements of ${\mathbb{Z}}_3\left[x\right]/\left(p(x)\right)$ and construct addition and multiplication tables for this field, where $p(x)=x^2+2$.