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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,

a, a^2

b, b^2

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c, c^2

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.

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The simplest basis I can think of consists of monomials $\mathcal{B}=\{1,a,a^2,b,ab,a^2b\}$. One way to see this goes as follows. Let’s first write

$$

P(x)=(x-a)(x-b)(x-c)=x^3-s_1x^2+s_2x-s_3,

$$

with the elementary symmetric polynomials $s_1=a+b+c$, $s_2=ab+bc+ac$ and $s_3=abc$. If $K$ is your base field, I claim that all the monomials $a^ib^jc^k$ can be written as linear combinations of monomials from the set $\mathcal{B}$ with coefficients from the field $L=K(s_1,s_2,s_3)$ (in your case surely $s_1,s_2,s_3\in K$, so the combinations are really $K$-linear).

The first and most obvious thing is to replace everywhere $c$ with $s_1-a-b$. After that we only need to take care of monomials $a^ib^j$. The quantities $a$ and $b$ are zeros of $P(x)$, so we know how to replace $a^i, i\ge 3$ and $b^j, j\ge3$ with lower powers. So we are left with the 9 monomials $\mathcal{B}\cup\{b^2,ab^2,a^2b^2\}$. Substituting $c=s_1-a-b$ to the equation $ab+ac+bc-s_2=0$ gives us a relation

$$

b^2=ab+a(s_1-a-b)+b(s_1-a)-s_2

$$

that allows us to write $b^2$ as an $L$-linear combination of $a^2,ab,a,b$ and $1$. When we do this substitution to a monomial like $ab^2$ or $a^2b^2$ we introduce higher powers of $a$. But we can reduce these to lower powers of $a$ as before. It may feel like $s_3$ wasn’t really used, but it did make an appearance, when we reduced the higher powers of $a$ and $b$.

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