Problem
Let A be the matrix
$\Bigg(\begin{matrix} 0&0&1\\ 1&0&0 \\0&1&0 \end{matrix} \Bigg)$
Giving brief justifications, determine whether A is diagonalizable over (a) the complex field; (b) the field $Z_2$ with two elements; (c) the field $Z_3$ with three elements; (d) the field $Z_7$ with 7 elements.
Progress
The characteristic polynomial $\chi_A(x)=x^3-1$, which in this case is equal to the minimum polynomial, $m_A(x)$. We make use of the fact that A is diagonalisable $\Leftrightarrow$ $m_A(x)$ can be expressed as the product of disjoint linear factors.
(a) In $\mathbb{C}$, $m_A(x)=x^3-1=(x-1)(x+\alpha)(x-\beta)$, where $\alpha \neq \beta$ and so A is diagonalisable.
Not sure how to apply the argument to the fields $\mathbb{Z}_n$ for $n=2,3,7$ however. Any help would be appreciated. Regards.