Apparently this question is asked twice or more on MSE, for example here, but I struggle to get the answers.
In The Geometry of Schemes by Eisenbud and Harris,exercise II-20, I am asked to classify all schemes of degree 3 over $\mathbb R$ supported at the origin in $\mathbb A_\mathbb R^2$. (That is, schemes of the form $\text{Spec } \mathbb R[x_1,\ldots, x_n ] /I$.) There is a hint that there are two nonisomorphic schemes in the list whose complexification is $\text{Spec } \mathbb C[x,y]/(x^2,y^2 ,xy).$
I am struggling to see why algebraic closure of $\mathbb C$ and algebraic non-closure of $\mathbb R$ is so important here. Apparently, for the same question over $\mathbb C$, I can just consider the vector space dimension of $\mathbb C[x_1,\ldots,x_n]/ I$ for an ideal $I$ and reach the conclusion that $I$ can take only one of a few forms. In this process, algebraic completenes doesn't seem to be used. To be more specific, if $\mathbb C[x_1,\ldots,x_n]/ I$ has dimension $3,$ then any polynomial $P$ in $\mathbb C[x_1,\ldots,x_n]$ can be written in the form $$ P = \sum_{1\leq i\leq 3}\lambda_i P_i + b, \lambda_i \in \mathbb R, b \in I, $$ where $P_i$ are a "basis". The expansion is easily seen to be unique. So by expanding the polynomials $1,x_i$ and so on, we will see that we can simplify the basis $P_i$ into simple polynomials $1,x_1,$ etc. Hence the particular simple schemes we are looking for. Where is algebraic closure used? Why doesn't this work for $\mathbb R$?
And what is the list of schemes we are looking for?