# Cyclo(X) = Cyclotomic polynomial of integer > 0

Cyclo(X) = Cyclotomic polynomial of integer > 0:

The cyclotomic polynomials Cyclo(n) for n = 1, 2, 3, . . . are the minimal polynomials for the primitive n-th roots of unity:

Cyclo(n) = Product{1 <= k <= n and GCD(k, n) = 1} [x − exp(2*pi*i*k/n)].

Cyclo(n) has degree phi(n), where phi signifies Euler's totient function. The first few are easily calculated to be x − 1, x + 1, x^2 + x + 1, x^2 + 1. All of the coefficients are −1, 0, or +1 for n < 105. For n > 1, the coefficients are symmetrical i.e. PolRecip(Cyclo(n)) = Cyclo(n). For all n, all of the coefficients are integers.

Cyclo(105) = x^48 + x^47 + x^46 − x^43 − x^42 − 2*x^41 − x^40 − x^39 + x^36 + x^35 + x^34 + x^33 + x^32 + x^31 − x^28 − x^26 − x^24 − x^22 − x^20 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 − x^9 − x^8 − 2*x^7 − x^6 − x^5 + x^2 + x + 1.

See: Cyclotomic Polynomial -- From Mathwords
See: Roots of unity -- From Wikipedia
See: On the Middle Coefficient of a Cyclotomic Polynomial -- From Professor Dresden