Cyclotomic polynomial factors

TLDR: The n th roots of unity play a key role in abstract algebra, providing a rich link between groups, vectors, regular n-gons, and algebraic factorizations.

Abstract: The n th roots of unity play a key role in abstract algebra, providing a rich link between groups, vectors, regular n-gons, and algebraic factorizations. This richness permits extensive study. A historical example of this interest is the 1938 challenge levelled by the Soviet mathematician N. G. Chebotarëv (see [1] or [2]). His question was ‘Are the coefficients of the irreducible factors in Z[n] of xn − 1 always from the set {−1, 0, 1}?’ Massive tables of data were compiled, but attempts to prove the results for all n failed. Three years later, V. Ivanov [3] proved that all polynomials xn 1, where n < 105, had the property that when fully factored over the integers all coefficients were in the set {−1, 0, 1}. However, one of the factors of x105 − 1 contains two coefficients that are −2. Ivanov further proved for which n such factorisations would occur and which term in the factor would have the anomalous coefficients. A twist that makes this historical episode more intriguing is that Bloom credited Bang with making this discovery in 1895, predating the Chebotarëv challenge by more than four decades.