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) = 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
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