** suivant:** Sturm sequences and number
** monter:** Arithmetic and polynomials
** précédent:** Chinese remainders : chinrem
** Table des matières**
** Index**

##

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

** suivant:** Sturm sequences and number
** monter:** Arithmetic and polynomials
** précédent:** Chinese remainders : chinrem
** Table des matières**
** Index**
giac documentation written by Renée De Graeve and Bernard Parisse