TITLE: Maximal ideals of polynomial rings in infinitely many variables
QUESTION [12 upvotes]: Let $k$ be an algebraically closed field. Nullstellensatz states that the maximal ideals of the polynomial ring $R=k[X_1,\dots,X_n]$ are precisely those of the form $\langle X_1-a_1,\dots,X_n-a_n\rangle$, with $(a_1,\dots,a_n)\in k^n$.

What if we are working with infinitely many variables, say $\{X_i\}_{i\in I}$, being $I$ an infinite set? Again, the ideals of the form $\bigl\langle\{X_i-a_i: i\in I\}\bigr\rangle$, with $(a_i)_{i\in I}\subseteq k$ are maximal (even if $k$ is not algebraically closed), but what about the converse?

REPLY [6 votes]: If $|k| > |I|$, then any maximal ideal of $k[\{x_i\}_{i \in I}]$ is of the standard form, see MO/41262. The "cheap proof" of the Nullstellensatz works. The example by Mariano shows that this doesn't hold if $|k|=|I|$.