Solution
= Solution
\b[Yes.] In $\mathbb F_7$, the <minimal polynomial> splits as
$$
X^3-1=(X-1)(X-2)(X-4).
$$
The three factors are distinct, so the <minimal polynomial criterion for diagonalizability> applies. For example the <eigenvectors> $v_\lambda=(1,\lambda^2,\lambda)^{\mathsf T}$ for $\lambda=1,2,4$ give an <eigenbasis> over this <finite field>. The contrast with characteristic three is that the shift polynomial now splits without repeated roots.