Cyclotomic polynomial : cyclotomic

cyclotomic takes an integer n as argument and returns the list of the coefficients of the cyclotomic polynomial of index n. This is the polynomial having the n-th pritmitive roots of the unity as zeros (a n-th root of the unity is primitive if the set of its powers is the set of all the n-th root of the unity).

For example, let n = 4, the fourth roots of the unity are: {1, i, -1, - i} and the primitive roots are: {i, - i}. Hence, the cyclotomic polynomial of index 4 is (x - i).(x + i) = x² + 1. Verification:

cyclotomic(4)

Output :

[1,0,1]

Another example, input :

cyclotomic(5)

Output :

[1,1,1,1,1]

Hence, the cyclotomic polynomial of index 5 is x⁴ + x³ + x² + x + 1 which divides x⁵ - 1 since (x - 1)*(x⁴ + x³ + x² + x + 1) = x⁵ - 1.
Input :

cyclotomic(10)

Output :

[1,-1,1,-1,1]

Hence, the cyclotomic polynomial of index 10 is x⁴ - x³ + x² - x + 1 and

(x⁵ -1)*(x + 1)*(x⁴ - x³ + x² - x + 1) = x¹⁰ - 1

Input :

cyclotomic(20)

Output :

[1,0,-1,0,1,0,-1,0,1]

Hence, the cyclotomic polynomial of index 20 is x⁸ - x⁶ + x⁴ - x² + 1 and

(x¹⁰ -1)*(x² +1)*(x⁸ - x⁶ + x⁴ - x² +1) = x²⁰ - 1