# How to compute the disciminant of QQ(\sqrt[3]{m}) using Sage's symbolic computations capabilities

R = QQ['m']    # m here is a variable
m = R.gen()

x = polygen(R) # a variable over R

A = R.extension(x^3 - m, 'a')  # a here will be \sqrt[3]{m}
a = A.gen()

# I define the trace pairing function
def trace_pairing(ring, elements):
    k = len(elements)
    T = matrix(R,k,k)   # an k by k matrix over R
    for [i, j] in CartesianProduct(range(0,k), range(0,k)):
        T[i,j] = (ring(elements[i]) * ring(elements[j])).trace() # coerces the elements to a ring with trace
    return T

# if m equiv 1 mod 9 then an integral basis is given by 1, a, (1+a+a^2)/3

T = trace_pairing(A, [1, a, (1+a+a^2)*(1/3)])    # also must do () * (1/3) instead of ()/3 because of a quirk in Sage
                                         # whereby ()/3 thinks of 3 in A which is not a field, whereas ()*(1/3) is just ring multiplication
T.det()