Cyclotomic polynomial : cyclotomic

### 2.25.14 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 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