### 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*)=*x*^{2}+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 *x*^{4}+*x*^{3}+*x*^{2}+*x*+1
which divides *x*^{5}−1 since (*x*−1)*(*x*^{4}+*x*^{3}+*x*^{2}+*x*+1)=*x*^{5}−1.

Input :

`cyclotomic(10)`

Output :

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

Hence, the cyclotomic polynomial of index 10 is *x*^{4}−*x*^{3}+*x*^{2}−*x*+1 and

(*x*^{5}−1)*(*x*+1)*(*x*^{4}−*x*^{3}+*x*^{2}−*x*+1)=*x*^{10}−1 |

Input :

`cyclotomic(20)`

Output :

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

Hence, the cyclotomic polynomial of index 20 is *x*^{8}−*x*^{6}+*x*^{4}−*x*^{2}+1 and

(*x*^{10}−1)*(*x*^{2}+1)*(*x*^{8}−*x*^{6}+*x*^{4}−*x*^{2}+1)=*x*^{20}−1 |