Maximal ideals in univariate polynomial rings $R[X]$ have a nice characterization in that they all are of the form $(E)$, for some irreducible $E\in R[X]$. This allows for a systematic way to construct maximal ideals in this setting.
I'm looking to do the same for multivariate polynomial rings. Let $k$ be a field (not algebraically closed — imagine that it's $\mathbb{F}_p$ for some prime $p$), and let $k[x_1,\ldots,x_n]$ be its polynomial ring in $n$ variables.
More specifically, I'm looking for maximal ideals $I\subseteq k[x_1,\ldots,x_n]$ such that for any polynomial $f\in k[x_1,\ldots,x_n]$ of total degree at most $d$, $f$ is the unique degree $\leq d$ polynomial such that $f = f \mod I$. Otherwise, there exists a degree $\leq d$ polynomial $g$ such that $g = f \mod I$. Intuitively, I would like to have something that behaves like modding by an irreducible $E$ in a univariate polynomial ring: if $E$ is of degree $d+1$, then $f \mod E$ has the behavior I described.
Any other pointers to examples/characterizations of maximal ideals in multivariate polynomial rings would be appreciated!
Thank you!
Best Answer
If each $p_i$ is irreducible, then the quotient $k[x_1, ... x_n]/(p_1, ... p_n)$ is canonically isomorphic to the tensor product of fields $\bigotimes k[x_i]/p_i$, so the question is when a tensor product of fields remains a field.
This seems to be a somewhat delicate field-theoretic question in general. Restricting to the case $n = 2$ for simplicity and writing $k_i = k[x_i]/p_i$, note that $k_1 \otimes k_2 \cong k_1[x_2]/p_2$, hence the question is whether $p_2$ remains irreducible when regarded as a polynomial over $k_1$. Then of course one has to repeatedly answer this question for each of the $p_i$.
I think a sufficient condition is that the pairwise intersection of the normal closures of the $k[x_i]/p_i$ in $\bar{k}$ is $k$.(This is wrong; see QiL's comment for the correct condition.) For example one might take $k = \mathbb{Q}$ and $p_i = x_i^2 - q_i$ where $q_i$ is an enumeration of the primes.