### 5.28.14  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 primitive roots of unity as zeros (an n-th root of unity is primitive if the set of its powers is the set of all the n-th roots of unity).

For example, let n=4, the fourth roots of unity are: { 1,i,−1,−i} and the primitive roots are: {i,−i}. Hence, the cyclotomic polynomial of index 4 is (xi).(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 x4x3+x2x+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 x8x6+x4x2+1 and

 (x10−1)*(x2+1)*(x8−x6+x4−x2+1)=x20−1