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.

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- April 19th 2010, 09:14 AMmathman88Varieties
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. - April 19th 2010, 10:41 AMNonCommAlg
- April 19th 2010, 12:57 PMmathman88
- April 19th 2010, 07:24 PMmathman88
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?

Thanks! - April 19th 2010, 08:54 PMchiph588@
- April 20th 2010, 09:34 AMmathman88
Thanks!

Now what about that field where is reducible? I'm having trouble picturing what that would be. - April 20th 2010, 10:43 PMchiph588@
I think will do the trick.