
  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!

For $k[x, y]$ this is not as bad as it sounds! The saving grace here is that $k[x]$ is a PID.

Proposition: Let $R$ be a PID. The prime ideals of $R[y]$ are precisely the ideals of the following form:


$(0)$,
$(f(y))$ where $f$ is an irreducible polynomial (recall that Gauss' lemma is valid over a UFD, so irreducibility over $R$ is equivalent to irreducibility over $\text{Frac}(R)$), 
$(p, f(y))$ where $p \in R$ is prime and $f(y)$ is irreducible in $(R/p)[y]$.


This is a nice exercise. If you get stuck, I prove it in in this blog post. The primes of the third type are maximal, so when $R = k[x]$ you've already listed them (by the weak Nullstellensatz). The only new prime ideals are those of the second type; they correspond to irreducible subvarieties of dimension $1$.

In general I'm not even sure what would count as a reasonable description, and I don't know enough algebraic geometry to comment. 
