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Cyclotomic polynomial factors

Published online by Cambridge University Press:  01 August 2016

Richard Grassl  and
Tabitha T.Y. Mingus
Richard Grassl
Affiliation:
University of Northern Colorado, Greeley, CO 80639, e-mail: Richard.Grassl@unco.edu
Tabitha T.Y. Mingus
Affiliation:
Western Michigan University, Kalamazoo, MI 49008, e-mail: Tabitha.Mingus@wmich.edu
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Extract

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.

Type
Articles
Information
The Mathematical Gazette , Volume 89 , Issue 515 , July 2005 , pp. 195 - 201
Copyright
Copyright © The Mathematical Association 2005

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References

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