Find out the degree of a polynomial $f(z)$ Let $\alpha$ and $\beta$ be two complex number with $\beta$ non-zero and $f(z)$ is a polynomial function on $\mathbb{C}$ such that $f(z)=\alpha$ whenever $z^{5}=\beta$ then what is degree of $f(z)?$  My guess is deg$f$ is $5$. But what is the right approach. 
Thanks in advance

By the fundamental theorem of algebra, $z^5 = \beta$ has exactly 5 solutions $\zeta_1,\ldots,\zeta_5$. Since $f$ is a polynomial, $f(\zeta_k) = \alpha$ is equivalent to $(z-\zeta_k)$ being a factor of $f-\alpha$, hence
$$ f(z) - \alpha = (z-\zeta_1)\cdot\ldots\cdot(z-\zeta_5)\cdot g(z)  $$
for some polynomial function $g$. Therefore all we can say is
$$\deg f = \deg \Big((z-\zeta_1)\cdot\ldots\cdot(z-\zeta_5)\cdot g(z) + \alpha\Big) \ge 5 $$
