This is about this question. For $n\in\mathbb{N}$ you can ask about how many nonisomorphic $n$-dimensional commutative $k$-algebras with only one prime ideal are there, where $k$ is an algebraically closed field. In that question we conclude that:
- For $n=2$ there is exactly one: $k[x]/(x^2)$.
- For $n=3$ there are at least two: $k[x]/(x^3)$ and $k[x,y]/(x^2,xy,y^2)$.
The argument also for $n=3$ works for $n\geq 3$. Take $S$ as the polynomial ring in $n-1$ variables and $I$ the ideal generated by all the monomials of degree $2$. Then $S/I$ is such an algebra with maximal ideal $(x,y)/I$.
So my question is... is there any results about how many nonisomorphic $k-$algebras of this type are there in each dimension? Are there finitely many in each dimension or is there some dimension with infinitely many ones?
My guess is... when $n\geq 6$ we can start making some constructions like $k[x,y]/I$ where $I$ is generated by the monomials of degree $3$ and since $\text{rad} I$ is also maximal this would be an algebra of this type (which is not isomorphic to $k[x]/(x^6)$ but a priori it could be isomorphic to $k[x_1,x_2,x_3,x_4,x_5]/J$ where $J$ is the ideal generated by the monomials of degree $2$), so I expect more and more of them will start appearing for bigger values of $n$.
