= Solution
The <minimal polynomial> $m_\alpha$ is the unique monic polynomial of least positive degree such that $m_\alpha(\alpha)=0$. The <Cayley-Hamilton theorem> says that $\alpha$ satisfies its characteristic polynomial, so such polynomials exist. Dividing any two monic candidates of least degree shows uniqueness.
If $m_\alpha(x)=x^m$, then $\alpha^m=0$ but $\alpha^{m-1}\ne0$. The least $r$ for which $(\alpha^3)^r=0$ is the least integer satisfying $3r\geq m$. Hence
$$
\boxed{m_{\alpha^3}(x)=x^{\lceil m/3\rceil}}.
$$
Back to article page
