Let $K$ be a cubic field such that $K=\mathbb Q[x]$ with $x^3=2$.
The discriminant of $\{1,x,x^2\}$ is supposed to be $\begin{vmatrix} 3 & 0 & 0 \\ 0 & 0 & 6 \\ 0 & 6 & 0\end{vmatrix}$, but when computing it with traces ($D(1,x,x^2)=\det(\text{Tr}_{K/ \mathbb Q}(x^ix^j)), i,j \in \{0,1,2\}$) I get $\begin{vmatrix} 3 & 3x & 3x^2 \\ 3x & 3x^2 & 6 \\ 3x^2 & 6 & 6x\end{vmatrix}$. What am I doing wrong?
(I have little experience with computing discriminants so far)