Let be the -axis contained in ( ).
In lecture, my professor said is irreducible (which makes sense), but I don't know how to show it. One hint he gave us is that "a prime ideal is radical" i.e. if is prime then .
Any insight is much appreciated.
So in general, here's what I'm thinking:
We have that is an algebraically closed field.
Now is irreducible is prime. But by Hilbert's Nullstellensatz we see that .
So is irreducible is prime.
I'm stuck after this. What is explicitly? Also I'm able to show if is a prime ideal then , but does the other direction work too?