TITLE: What do prime ideals in $k[x,y]$ look like?
QUESTION [65 upvotes]: Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like?

As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$ where $a,b\in k$. What can we say about the prime ideals? Are there similar results? And what about $k[x,y,z], k[x,y,z,w]$ and so on. Would someone be kind enough to give me some hints or referrence on this topic? Thank you very much!

REPLY [4 votes]: For $k[x,y]$ there are excellent answers here; however, if one still wonders what happens in $k[x_1,\ldots,x_n]$, $n>2$ (as was commented),
then it is worth to mention Theorem 2.6 of
Miguel Ferrero's paper,
which deals with prime ideals in polynomial rings in several variables, $n \geq 2$.