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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 (xi).(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 x4x3+x2x+1 and

(x5−1)*(x+1)*(x4x3+x2x+1)=x10−1 

Input :

cyclotomic(20)

Output :

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

Hence, the cyclotomic polynomial of index 20 is x8x6+x4x2+1 and

(x10−1)*(x2+1)*(x8x6+x4x2+1)=x20−1 

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