Prime Ideals: Q1, Q2, Q3

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In summary, the conversation discusses whether certain sets of polynomials form prime ideals in the ring C[x1, x2, x3, x4], with the focus on the twisted cubic in projective 3-space. The conclusion is that the ideal (x1, x2) is prime, while the ideal (x1 x4-x2 x3, x1 x3-x2^2) is not. However, the ideal (x1 x4-x2 x3, x1 x3-x22, x2 x4-x32) is prime due to the irreducibility of the twisted cubic. There is also discussion about the need for a proper proof and the difficulty of decomposing ideals.
  • #1
naturemath
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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.
 
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  • #2
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.
 
  • #3
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?
 
  • #4
math2012 said:
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.
 
  • #5
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.
 
  • #6
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.
 
  • #7
> 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.
 
  • #8
> 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?
 
  • #9
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.
 
  • #10
Oh I see. Thank you mathwonk!
 

Q1: What is a prime ideal?

A prime ideal is a type of ideal in a ring that has special properties. It is a subset of the ring that is closed under multiplication and addition, and it is also an ideal that cannot be further factored into smaller ideals.

Q2: How are prime ideals different from other types of ideals?

Unlike other types of ideals, prime ideals have the property that if the product of two elements in the ideal is in the ideal, then at least one of the elements must also be in the ideal.

Q3: What is the importance of prime ideals in mathematics?

Prime ideals are important in many areas of mathematics, including number theory, algebraic geometry, and commutative algebra. They play a crucial role in studying the properties of rings and fields, and they help to classify and understand various mathematical structures.

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