CoCalc Shared Filesminimal polynomial zeta_m .sagewsOpen in CoCalc with one click!
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tab = [4,6,8,9,10,12] n = 3 for m in tab : l = lcm(n,m) K.<w> = CyclotomicField(l) R.<x> = K[] P = 1 a = w^(l/m) for d in [1..l/n] : k = n*d+1 if gcd(l,k) == 1 : P = P * (x-a^k) [m,P]
[4, x^2 + 1] [6, x - w] [8, x^4 + 1] [9, x^3 - w^3] [10, x^4 - x^3 + x^2 - x + 1] [12, x^2 - w^2]
S.<x,t> = QQ[] tab = [4,6,8,9,10,12] #tab = [4] n = 3 for m in tab : l = lcm(n,m) P = 1 a = t^(l/m) for d in [1..l/n] : k = n*d+1 if gcd(l,k) == 1 : P = P * (x-a^k) K.<w> = CyclotomicField(l) p = P(x,w) [m,p]
[4, x^2 + 1] [6, x + (-zeta6)] [8, x^4 + 1] [9, x^3 + (-zeta9^3)] [10, x^4 - x^3 + x^2 - x + 1] [12, x^2 + (-zeta12^2)]
K.<c> = CyclotomicField(3) S.<x> = K[] R.<w> = S[] I = R.ideal(w^4-t) H.<W> = R.quotient(I) (x-W^7)*(x-W)
-x*t*W^3 - x*W + x^2 + t^2