The field with eight elements

#ring_theory

Let $Z_{2} = {0, 1}$ be the field of two elements. The quotient ring $Z_{2} [x] / ⟨ x^{3} + x + 1 ⟩$ is a field of cardinality 8, containing $Z_{2}$ . Let $π : Z_{2} [x] \to Z_{2} [x] / ⟨ x^{3} + x + 1 ⟩$ be the natural projection.

Write down a set of eight distinct coset representatives for the elements of this field.
Determine the multiplicative inverse of $π (x)$ in terms of your coset representatives.

View

L A T E X

code

Let ${\bf Z}_2=\{0,1\}$ be the field of two elements. The quotient ring ${\bf Z}_2[x]/\langle x^3+x+1\rangle$ is a field of cardinality 8, containing ${\bf Z}_2$. Let $\pi:{\bf Z}_2[x]\to {\bf Z}_2[x]/\langle x^3+x+1\rangle$ be the natural projection.
\begin{enumerate}[label=\alph*)]
	\item Write down a set of eight distinct coset representatives for the elements of this field.
	\item Determine the multiplicative inverse of $\pi(x)$ in terms of your coset representatives.
\end{enumerate}