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## 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