= Solution
\b[No.] The same cyclic-vector argument gives <minimal polynomial> $X^3-1=(X-1)(X^2+X+1)$ over the <finite field> $\mathbb F_2$. The quadratic has no root in $\mathbb F_2$: its value is one at both zero and one. Thus the <minimal polynomial> does not split into linear factors over the specified <field>, violating the <minimal polynomial criterion for diagonalizability>. It becomes diagonalizable over a suitable larger <finite field>, but that is a different question.
Back to article page
