I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it might be good to do some computations to understand how this theory works by taking a concrete example.
Given a cubic field I want to find the corresponding binary cubic form associated with it or rather with the ring of integer of $K$ should I say?
Suppose we take $$K=\mathbb{Q}(\sqrt[3]{2})\supset\mathbb{Q}$$
Since $$x^{3}-2\in\mathbb{Q}[x]$$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion with $p=2$ we get $$[K:\mathbb{Q}]=3$$
The ring of integers of $K$ is $$\mathcal{O}_{k}=\mathbb{Z}[\alpha]=\mathbb{Z}[\sqrt[3]{2}]$$ with integral basis $\{1,\alpha,\alpha^2\}$ hence $$\mathbb{Z}[\sqrt[3]{2}]=\{a+b\sqrt[3]{2}+c\sqrt[3]{2^2}|a,b,c\in\mathbb{Z}\}$$
In the next step I would like to find (if possible) the corresponding binary cubic form. Will this form correspond to the field of to the ring?
I follow Belabas and Cohen ''Binary cubic forms and cubic number fields'': Let $K$ be number field defined by a root $\theta$ of the polynomial $x^3+px^2+qx+r$ with $p,q,r \in\mathbb{Z}$ such that there exists an integral basis of the form $(1,\theta,(\theta^2+t\theta+u)/f)$ with $t,u,f\in \mathbb{Z}$ and $f=[\mathbb{Z}_{k}:\mathbb{Z}[\theta]]$ Then we choose $\alpha=\theta$ and $\beta=(\theta^2+t\theta+u)/f)$ we have explicitly: $$\begin{split} F_{B}(x,y)&=((t^3-2t^2p+t(q+p^2)+r-pq)/f^2)x^3\\&+((-3t^2+4tp-(p^2+q))/f))x^2y\\ &+(3t-2p)xy^2-fy^3\end{split}$$
In case of $x^{3}-2$ we have $\begin{cases}q=0\\r=-2\\p=0\end{cases}$
And $f=[\mathbb{Z}_{k}:\mathbb{Z}[\theta]]=1$
Plugging everything in I get something like this $F_{B}(x,y)=-2x^3-y^3$
Does this work? Or is it completely incorrect? $f=[\mathbb{Z}_{k}:\mathbb{Z}[\theta]]$ shouldn't this expression give index form? (i.e binary cubic form?)
Thank you.
