minimal polynomial zeta_m .sagews

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