I'm trying to express the splitting field of a cubic equation as a vector space over the rationals. Specifically I am looking for a set of six independent vectors that span the space. If the roots are a,b, and c then I know I need to be able to express all expressions in a,b and c up to second degree: namely,
ab, bc, ca
a^2b, b^2c, c^2a, ab^2, bc^2, ca^2
Obviously there are 15 of these and they are not all independent. Given a and b, I easily get c as a linear term in a, b, and (a+b+c). Likewise c^2 is not independent of a^2 and b^2. And I find that I can generate ab etc as linear combinations of c, c^2, and abc.
So what are my six vectors? I have allowed myself 1, a, b, a^2, and b^2 ....this gives me five and I think they are all linearly independent. I'm only allowed one more and I can't get it to work. I'm inclined to try (a^2b + b^2c + c^2a) because I'm pretty sure I need it, but having done so I can't see how I generate terms like a^2b on their own.
Any ideas? I'd be especially interesteds if there is a basis which is more symmetric in some sense than the arbitrary collection of terms which I'm cobbling together.