Let $\alpha,\beta $ be two complex numbers with $\beta \neq 0$ and $f(z)$ be a polynomial function on $\Bbb C$ such that $f(z)=\alpha $ whenever $z^5=\beta $.What can you say about degree of $f(z)$?
My effort: I am unable to find the number of roots of $z^5-\beta =0$ .As $\Bbb C$ is algebraically closed so any polynomial would split and hence $z^5-\beta=0$ has exactly $5$ roots counting multiplicity.
So $f-\alpha $ has degree atleast $5$ if $z^5-\beta =0$ has distinct $5$ roots
Is it correct?I am unable to provide a specific answer.
Please give some hint.