In mathematics, the cyclotomic identity states that Mathematics is the study of quantity, structure, space and change. ...
where M is Moreau's necklace-counting function In combinatorial mathematics, Moreaus necklace-counting function where μ is the classic Möbius function, counts the number of necklaces asymmetric under rotation (also called Lyndon words) that can be made by arranging n beads the color of each of which is chosen from a list of α colors. ...
and μ is the classic Möbius function of number theory. The denominator on the right, 1 − zj, is a cyclotomic polynomial -- hence the name. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
Reference
Nicholas Metropolis & Gian-Carlo Rota. The Cyclotomic Identity. Reprinted in Gian-Carlo Rota on Combinatorics. Birkhäuser. Boston. 1995. Nicholas Constantine Metropolis (June 11, 1915 - October 17, 1999) was a mathematician and physicist. ... Gian-Carlo Rota (April 27, 1932 – April 18, 1999, known as Juan Carlos Rota to Spanish speakers) was an Italian-born American mathematician and philosopher. ...
− 1; the primitive n-th roots of unity are precisely the zeros of the nth cyclotomic polynomial
This field contains all nth roots of unity and is the splitting field of the nth cyclotomic polynomial over Q.
Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.
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