What can be said about the structure of maximal ideals of $R=k[\{x\_i\}\_{i \in I}]$, or geometric properties of $\text{Spm } k[\{x\_i\}\_{i \in I}]$? Here $k$ is an arbitrary field and $I$ is an *infinite* set. Kernels of evaluation homomorphisms yield an injective map


$\overline{k}^I / Aut(\overline{k}/k) \to \text{Spm } k[\{x\_i\}\_{i \in I}]$.


The image consists of those maximal ideals whose residue field is algebraic over $k$. If $I$ is finite, every residue field is algebraic (Noether Normalization). However, if $I$ is infinite and $|I| \geq |k|$, for example $k(t)$ is a residue field which is not algebraic. What happens if $|k| > |I|$? Is there a description in the general case?



maximal ideals of $k[x_1,x_2,...]$
If $|k| > |I|$ then the usual cheap proof of Nullstellensatz still works: let $K$ be a residue field. Then $\dim\_k K \le \dim\_kR = |I|$, but if $t\in K$ is transcendental over $k$, the elements $1/(t-a)$ for $a\in k$ are $k$-linearly independent. So $K/k$ is algebraic.


