2013-11 Integer coefficient complex-valued polynomials

Determine all polynomials $$P(z)$$ with integer coefficients such that, for any complex number $$z$$ with $$|z| = 1$$, $$| P(z) | \leq 2$$.

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Let $$p$$ be a prime number. Let $$S_p$$ be the set of all positive integers $$n$$ satisfying
$x^n – 1 = (x^p – x + 1) f(x) + p g(x)$
for some polynomials $$f$$ and $$g$$ with integer coefficients. Find all $$p$$ for which $$p^p -1$$ is the minimum of $$S_p$$.