The field with nine elements

#ring_theory

Let $F_{9}$ denote the field of nine elements.
(a) Show that each nonzero $a \in F_{9}$ is a root of $X^{8} - 1 = (X - 1) (X + 1) (X^{2} + 1) (X^{4} + 1) \in F_{3} [X]$ .
(b) Use the Pigeonhole Principle to prove that $F_{9}$ has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)

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Let ${\bf F}_9$ denote the field of nine elements.
\begin{enumerate}[label=\alph*)]
	\item Show that each nonzero $a\in {\bf F}_9$ is a root of $X^8-1=(X-1)(X+1)(X^2+1)(X^4+1)\in {\bf F}_3[X]$.
	\item Use the Pigeonhole Principle to prove that ${\bf F}_9$ has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)
\end{enumerate}