We suggest a construction of the minimal polynomial mΞ²kβ of
Ξ²kβFqnβ over Fqβ from the minimal polynomial f=mΞ²β for all positive integers k whose prime factors divide qβ1. The
computations of our construction are carried out in Fqβ. The key
observation leading to our construction is that for kβ£qβ1 holds
mΞ²kβ(Xk)=j=1βtkββΞΆkβjnβf(ΞΆkjβX),
where t=max{mβ£gcd(n,k):f(X)=g(Xm),gβFqβ[X]} and
ΞΆkβ is a primitive k-th root of unity in Fqβ. The
construction allows to construct a large number of irreducible polynomials over
Fqβ of the same degree. Since different applications require
different properties, this large number allows the selection of the candidates
with the desired properties