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).

  1. 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 $mathbb C[T]/(p) = mathbb C[X, Y]/mathfrak m$?

  2. More generally, What happens over $mathbb C[X_1, dots, X_n]$?

  3. Even more generally, over any algebraically closed field $F$?

  4. 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?

Add your own answers!

Related Questions

Proving that $(0,1)$ is uncountable

2  Asked on November 29, 2021 by henry-brown


Always factorise polynomials

1  Asked on November 29, 2021 by beblunt


Ask a Question

Get help from others!

© 2023 All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP