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Let $\alpha ,\beta$ be two complex numbers with $\beta \neq 0$ , 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 $f(z)$ ?

How can I solve the problem?

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How many roots do $f(z)-\alpha$ and $z^5-\beta$ have in common? – Hagen von Eitzen Sep 19 '12 at 17:41
@hum: What have you tried so far? – Michael Albanese Sep 26 '12 at 18:18

The solutions of $z^5=\beta$ are the same as the roots of $Z^5=1$ where $Z=\frac z{|\beta|^{1/5}}$, hence $z^5-\beta$ has five distinct roots. So $f-\alpha$ has at least five different roots, hence $f$ is a polynomial of degree at least $5$.

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