$\newcommand{\C}{\mathbb C} \newcommand{\m}{\mathfrak m}$ Let $\m$ be a maximal ideal of $\C[X, Y]$. We know that $\C[X, Y]/\m \simeq K$ will be a field (ring quotient maximal ideal is a field).
Is it true that we this field $K$ will always be algebraic over $\C$? That is, there exists some irreducible polynomial $p(t) \in \mathbb C[T]$ such that $\mathbb C[T]/(p) = \mathbb C[X, Y]/\mathfrak m$?
More generally, What happens over $\mathbb C[X_1, \dots, X_n]$?
Even more generally, over any algebraically closed field $F$?
Even more broadly, is there a theory of "multivariate field extensions"? When I learnt finite field theory from Artin, we only considered the case of $F[X]/\mathfrak m$ for a field $F$, maximal ideal $\m$. What is the broader theory of $F[X_1, \dots X_n] / \mathfrak m$? Can we "reduce" this multi-variate case to the single variable case, something like how we reduce multilinear algebra to linear algebra by facotring through the tensor product?