Determine the prime ideals of the polynomial ring C[x, y] in two variables

[Simfish]

So the problem is:
"4:(a) Determine the prime ideals of the polynomial ring C[x, y] in two variables."

"We recognize that an ideal P is prime if and only if for two ideals A and B, AB $\in$ P implies that either A or B is contained in P. So we must find "

So anyways, I'm thinking that it consists of all the irreducible polynomials in C[x,y] (I suppose those irreducibles can form ideals by means of multiples of those with other elements in the ring). (although we can't even categorize all the irreducibles in "

"Hm

First of all, standard irreducibility tests don't work
(because it's a complex domain, so (x^2 + 1) is reducible in this case). So then in C[X] at least we have polynomials of first degree that are irreducible..

So then we have to find irreducibles over complex numbers. BUT on the OTHER hand, we have xy, so now we can have irreducible factors of X and Y (maybe, irreducible polynomials can be elliptic curves like the one on Wikipedia)."

Y^2 - X^3 - X - 1 is prime ideal from wikipedia.

 

[morphism]

There are a couple of ways you can approach this. One is to think about when C[x,y]/I is going to be an integral domain, and the other is to think about algebraic varieties.