 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 ${\alpha }^{m-1}\ne 0$. The least $r$ for which $({\alpha }^3{)}^r=0$ is the least integer satisfying $3r\ge m$. Hence