Prime Ideals: Abstract Algebra Example

[naturemath]

This is a basic abstract algebra question.

Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ?

Q2. What about this: (x₁ x₄-x₂ x₃, x₁ x₃-x₂²)?

Q3. Is this a prime ideal (this is the twisted cubic in projective 3-space):
(x₁ x₄-x₂ x₃, x₁ x₃-x₂², x₂ x₄-x₃²)?

Thanks everyone.

 

[micromass]

What are your thoughts??

Hint: I always like to check if something is a prime ideal by checking if the quotient ring is an integral domain.

 

[math2012]

A1. Yes
A2. Yes
A3. No? Not sure.

-x2 (x1*x4-x2*x3)+x3 (x1*x3-x2^2)+x1 (x2*x4-x3^2)=0 (*)

and

-x3 (x1*x4-x2*x3)+x4 (x1*x3-x2^2)+x2 (x2*x4-x3^2)=0 (**)

Are (*) and (**) relevant at all?

 

[micromass]

A1. Yes
A2. Yes

Why yes to both??

A3. No? Not sure.

-x2 (x1*x4-x2*x3)+x3 (x1*x3-x2^2)+x1 (x2*x4-x3^2)=0 (*)

and

-x3 (x1*x4-x2*x3)+x4 (x1*x3-x2^2)+x2 (x2*x4-x3^2)=0 (**)

Are (*) and (**) relevant at all?

I don't see how (*) and (**) are relevant to this.

 

[naturemath]

So I'm guessing the following.

Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ?

Yes since the quotient is an integral domain (an irredu variety-- it's the x3 x4-plane).

Q2. What about this: (x1 x4-x2 x3, x1 x3-x2^2)?

No since
x2 (x1 x4-x2 x3) -x3 (x1 x3-x2^2) = x1(x2 x4-x3^2) is in the ideal but neither x1 nor (x2 x4-x3^2) is in the ideal.

Q3. Is this a prime ideal (this is the twisted cubic in projective 3-space):
(x1 x4-x2 x3, x1 x3-x22, x2 x4-x32)?

Yes, because this is the twisted cubic, which is irreducible.

 

[micromass]

So I'm guessing the following.

Q1. Is this (x1, x2) a prime ideal in C[x1, x2, x3, x4] ?

Yes since the quotient is an integral domain (an irredu variety-- it's the x3 x4-plane).

Q2. What about this: (x1 x4-x2 x3, x1 x3-x2^2)?

No since
x2 (x1 x4-x2 x3) -x3 (x1 x3-x2^2) = x1(x2 x4-x3^2) is in the ideal but neither x1 nor (x2 x4-x3^2) is in the ideal.

Yes, but strictly speaking you'll need to show that neither x1 nor (x2 x4-x3^2) is in the ideal.

Q3. Is this a prime ideal (this is the twisted cubic in projective 3-space):
(x1 x4-x2 x3, x1 x3-x22, x2 x4-x32)?

Yes, because this is the twisted cubic, which is irreducible.

This is hardly a proof.

 

[naturemath]

> Yes, but strictly speaking you'll need to show that neither x1 nor (x2 x4-x3^2) is in the ideal.

Thanks.

> This is hardly a proof.

Yes, but it seems quite messy to do it directly, using the polynomials.

 

[naturemath]

> This is hardly a proof.

I'm thinking of writing the ideal (or any ideal) as a primary decomposition, but for even that, is there a systematic way to decompose an ideal in such a way? Or do you recommend other (more) feasible options?

 

[mathwonk]

the tricky part is that many ideals have the same zero locus but at most one of those ideals is prime. in the case of the twisted cubic, if an ideal I has the twisted cubic as its zero locus, the irreducibility of the cubic implies the radical of I is prime, but not necessarily I itself. so you have to do the algebra.

 

[naturemath]

Oh I see. Thank you mathwonk!