Chapter 10, Problem 17E

To determine

To show: The polynomial $x^3+x+1$ cannot be factored in a nontrivial way with coefficients in $Z_2$ and use the polynomial to construct a field with $2^3=8$ elements.

(a)

To determine

The value of $\left(1+i\right)+\left(1+i+i^2\right)$.

(b)

To determine

The value of $\left(1+i^2\right)+\left(1+i^2\right)$.

(c)

To determine

The value of $i^{-1}$.

(d)

To determine

The value of $i^2\times \left(1+i+i^2\right)$.

(e)

To determine

The value of $\left(1+i\right)\left(1+i+i^2\right)$.

(f)

To determine

The value of ${\left(1+i\right)}^{-1}$.