Let $\mathbb K$ be a number field of degree $n$ over $\mathbb Q$, and let $\alpha_1,\alpha_2, \ldots ,\alpha_n$ be a $\mathbb Q$-basis of $\mathbb K$. Then there are coefficients $(c^{ij}_k)$ (where $i,j,k$ are independent indices between $1$ and $n$) such that
$ \alpha_i \times \alpha_j = \sum_{k=1}^{n} c^{ij}_k \alpha_k $
and we have for any indices $i,j,k,l$,
$ (1) c^{ij}_k=c^{ji}_k \ (\ {\rm commutativity}) $
$ (2) \sum_{y=1}^{n}c^{iy}_lc^{jk}_y=\sum_{y=1}^{n}c^{ij}_yc^{yk}_l \ (\ {\rm associativity}) $
If we take $\alpha_n$ to be $1-\sum_{y=1}^{n-1} \alpha_y$, we also have
$ (3) \sum_{y=1}^{n}c^{yi}_j=\delta_{ij} $
where $\delta_{ij}$ is the Kronecker symbol.
Now, let $V$ be the algebraic variety in the variables $( c^{ij}_k)$ defined as the subset of ${\mathbb C}^{n^3}$ satisfying equations (1) to (3). Is the dimension of $V$ (in the sense of algebraic geometry) known ?