$newcommand{C}{mathbb C} newcommand{m}{mathfrak m}$ 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?

Is a field extension of the form $mathbb C[X, Y]/mathfrak m$ isomorphic to $mathbb C(k)$ for some algebraic $k$?

Mathematics Asked on January 3, 2022

$newcommand{C}{mathbb C}
newcommand{m}{mathfrak m}$
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?

commutative algebra extension field field theory

Is a field extension of the form $mathbb C[X, Y]/mathfrak m$ isomorphic to $mathbb C(k)$ for some algebraic $k$?

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