Problem 2

Let $\alpha$ be a zero of $x^{3}+x^{2}+1$ in some extension field of $\mathbb{Z}_{2}$.

1. Find $(\alpha^{2}+\alpha+1)^{-1}$ in $\mathbb{Z}_{2}(\alpha)$.

2. Hence solve the equation

   $$(\alpha^{2}+\alpha+1)x+\alpha^{2}=\alpha$$

   for $x$ over $\mathbb{Z}_{2}(\alpha)$.

(Hint: refer to the conversion formulae given in 1.7.)