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 (x−i).(x+i)=x2+1. Verification:
Output :
Another example, input :
Output :
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 :
Output :
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 :
Output :
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 |