The field with nine elements
Let
- Show that each nonzero
is a root of . - Use the Pigeonhole Principle to prove that
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 $X^3-1=(X-1)(X^2+1)(X^4+1)\in {\bf F}_3[X]$.
\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}