We're given that $K/F$ is a Galois extension, and $[K:F]= n.$ We fix $\alpha \in K$ with minimal polynomial $m_{\alpha}$. Now, we define a linear transformation of F-vector spaces $\mu_{\alpha}: K \rightarrow K $ by the rule $\mu_{\alpha}(x) = \alpha x$. Show that for $f ∈ F[x]$ we have that $f(\mu_{\alpha}) = 0$ iff $f(x) \in (m_{\alpha}(x)).$
I was able to prove that if $f(\mu_{\alpha}) = 0$, then $f(x) \in (m_{\alpha}(x)).$ However, I am stuck with the converse part. Could you give me some hints? Thanks so much.
Notice that $$\underbrace{\mu_{\alpha}\circ\mu_{\alpha}\circ\cdots\circ\mu_{\alpha}}_{n\,\text{times}}=\mu_{\alpha^n}.$$ So, $m_{\alpha}(\mu_{\alpha})=\mu_{m_{\alpha}(\alpha)}=0.$ If $f=gm_{\alpha}$, then $f(\mu_{\alpha})=gm_{\alpha}(\mu_{\alpha})=0.$