Cyclotomic polynomials

circular polynomials

The polynomials that satisfy the relation

where the product is taken over all positive divisors of the number , including itself. Over the field of complex numbers one has

where ranges over the primitive -th roots of unity (cf. Primitive root). The degree of is the number of integers among that are relatively prime with . The polynomials can be computed recursively by dividing the polynomial by the product of all , , . The coefficients lie in the prime field ; in case of characteristic zero, they are integers. Thus,

If, moreover, is prime, then

The polynomial can be explicitly expressed using the Möbius function :

For example,

All the polynomials are irreducible over the field of rational numbers, but they may be reducible over finite prime fields. Thus, the relation

is valid over the field of residues modulo 11.

The equation , which gives all primitive -th roots of unity, is known as the equation of division of the circle. The solution of this equation in trigonometric form is

where the fraction is irreducible, i.e. and are relatively prime. The solution of the equation of division of the circle by radicals is closely connected with the problem of constructing a regular -gon, or with the equivalent problem of subdividing the circle into equal parts; the latter problem can be solved using a straightedge and a pair of compasses if and only if the equation is solvable in quadratic radicals. It was shown by C.F. Gauss in 1801 that this condition is satisfied if and only if

where is a non-negative integer and are pairwise different prime numbers of the form , where is a non-negative integer.

References

[1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2]	A.K. Sushkevich, "Foundations of higher algebra" , Moscow-Leningrad (1941) (In Russian)

Comments

The degree of is equal to , the value (at ) of the Euler -function (cf. Euler function).

Prime numbers of the form with a non-negative integer are called Fermat primes, these numbers are related to a problem of Fermat: When is the number , with as before, prime? The only small Fermat primes are , , , , (cf. [a1], pp. 183-185).

References

[a1]	I. Stewart, "Galois theory" , Chapman & Hall (1973)
[a2]	H. Davenport, "Multiplicative number theory" , Springer (1980)