Let $\alpha,\beta$ be two complex numbers with $\beta\ne 0$ and $f(z)$ a polynomial function on $\mathbb C$ such that $f(z)=\alpha$ whenever $z^5=\beta$. What can you say about the degree of the polynomial $f(z)$?
It is very clear that the degree of the polynomial $f(z)$ must be at least five. Can we say anything more specific about the degree? Please help.
It is clear that $f(z) = (z^5 - \beta) + \alpha$ works. Furthermore, this is the simplest monic polynomial such that $f(z) = \alpha$ if and only if $z^5 = \beta$. Others are $(z^5 - \beta)^n + \alpha$ for $n \ge 2$. From them you can manufacture polynomials of higher degrees with other zeros, e.g. $((z^5 - \beta)^4 + \alpha) \cdot (z - \gamma)$. So the answer is "any degree higher than 5".