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2011 PUMaC Algebra A7 / B8

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September 24, 2019
algebra

Problem Statement

Let α1,α2,…,α6\alpha_1,\alpha_2,\dots,\alpha_6 be a fixed labeling of the complex roots of x6−1x^6-1. Find the number of permutations {αi1,αi2,…,αi6}\{\alpha_{i_1},\alpha_{i_2},\dots,\alpha_{i_6}\} of these roots such that if P(α1,…,α6)=0P(\alpha_1, \dots, \alpha_6) = 0, then P(αi1,…,αi6)=0P(\alpha_{i_1},\dots,\alpha_{i_6}) = 0, where PP is any polynomial with rational coefficients.
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