Characterization of prime ideals of a ring of polynomials with coefficients in an algebraically closed field Is there any way to characterize the prime ideals of a ring of polynomials with coefficients in an algebraically closed field?
Observation: I believe that when taking $A = K[x,y]$ a ring of polynomials with $K$ algebraically closed field, as for example, we have the prime ideals of $A$ are all given by $(0)$, $(f)$, with $f$ irreducible to $A$, or $(x-a, y - b)$, with $a,b \in K$.
I am grateful for any reference that mentions a result that makes such a characterization.
 A: Let $K$ be any field. Let $p$ be a prime ideal of $K[x_1, \ldots, x_n]$ that is $\ne 0$. Let $0\ne f \in p$. Recall that $K[x_1, \ldots, x_n]$ is a unique factorization domain. Decompose $f$ into irreducible
$f= f_1 \ldots f_k$. Since $p$ is prime, one of the $f_i$ will be in $p$, say $f_1$. Now we have $(f_1) \subset p$.
What we showed is that prime ideal contains a principal prime ideal.
Now, prime ideals containing $(f_1)$ correspond to prime ideals of the domain $K[x_1, \ldots x_n]/(f_1)$. Now, Noether's normalization lemma says that for any finitely generated $K$ algebra that is a domain,  there exist $y_1$, $\ldots$, $y_l$ in $A$, algebraically independent, such that
$$K[y_1, \ldots, y_l]\subset A$$
is an integral extension.
Also, we know that there exists a correspondence between the prime of $B=K[x_1, \ldots, x_l]$ and the ones of an integral extension $A$ of $B$ ( for all this, Atiyah and Macdonald, Commutative algebra-- or many other books, are fine). We can see that we could do some inductive argument.
In any case, for $K[x,y]$ it means that the chains
$0 \subset (f) \subset m_{\alpha}$ are largest possible . Here $m_{\alpha}$ is the maximal ideal consisting of all the polynomials $f(x,y)$ such that $f(\alpha_1, \alpha_2) = 0$. Also, $\alpha= (\alpha_1, \alpha_2) \in \bar K ^2$, where $\bar K$ is the algebraic closure of $K$. Note that two $\alpha$ and $\beta$ define the same ideal if and only if there exists $\sigma \in \operatorname{Gal}( \bar K/K)$ such that $\sigma \alpha = \beta$.
As an example, we can consider the ideal of all polynomial in $\mathbb{Q}[x,y]$ such that $P(\sqrt{2} + \sqrt{3}, 1+ \sqrt{6}) = 0$. It is a maximal ideal. It can be also defined as the ideal of polynomials $P\in \mathbb{Q}[x,y]$, with $P(\sqrt{2}-\sqrt{3}, 1 - \sqrt{6}) = 0$.
