 Solution

Invariance of $W$ means every power of $\alpha$ restricts to the corresponding power of $\beta$. Therefore $m_{\alpha }(\beta )$ is the restriction of $m_{\alpha }(\alpha )$, and it is zero. Divide $m_{\alpha }=q{m}_{\beta }+r$ with $\mathrm{deg}r<\mathrm{deg}{m}_{\beta }$. Evaluation at $\beta$ gives $r(\beta )=0$. By minimality of $m_{\beta }$, $r=0$, provingThe zero-subspace case has $m_{\beta }=1$ and satisfies the same conclusion.