Problem

Source: Iran 3rd round 2013 - Algebra Exam - Problem 5

Tags: algebra, polynomial, trigonometry, complex numbers, algebra proposed



Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon. (30 points)