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Cyclotomic Equation

The equation

x^p=1,

where solutions zeta_k=e^(2piik/p) are the roots of unity sometimes called de Moivre numbers. Gauss showed that the cyclotomic equation can be reduced to solving a series of quadratic equations whenever p is a Fermat prime. Wantzel (1836) subsequently showed that this condition is not only sufficient, but also necessary. An "irreducible" cyclotomic equation is an expression of the form

(x^p-1)/(x-1)=x^(p-1)+x^(p-2)+...+1=0,

where p is prime. Its roots z_i satisfy |z_i|=1.

See also

Cyclotomic Polynomial, de Moivre Number, Polygon, Primitive Root of Unity

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References

Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 99-100, 1996.Scott, C. A. "The Binomial Equation x^p-1=0." Amer. J. Math. 8, 261-264, 1886.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 1, 366-372, 1836.

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Cyclotomic Equation

Cite this as:

Weisstein, Eric W. "Cyclotomic Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclotomicEquation.html

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