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non-zero roots in the unitary circumference

Source: China TST 1989, problem 3

June 27, 2005
Gausscalculusderivativealgebrapolynomialcombinatorial geometryChina

Problem Statement

Find the greatest nn such that (z+1)n=zn+1(z+1)^n = z^n + 1 has all its non-zero roots in the unitary circumference, e.g. (α+1)n=αn+1,α≠0(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0 implies ∣α∣=1.|\alpha| = 1.
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