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

In summary, the problem is to determine the prime ideals of the polynomial ring C[x,y] in two variables. To solve this, we need to find irreducible polynomials over complex numbers and consider when C[x,y]/I is an integral domain. Another approach is to think about algebraic varieties.
  • #1
Simfish
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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.
 
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  • #2
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.
 

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

1. What is a prime ideal?

A prime ideal is a subset of a ring (such as the polynomial ring C[x, y]) that has two main properties: it is closed under multiplication and it is an ideal. In simpler terms, it is a non-empty set of elements that, when multiplied together, always result in an element within the set. Additionally, it is also closed under addition and scalar multiplication.

2. How do you determine the prime ideals of a polynomial ring?

To determine the prime ideals of a polynomial ring C[x, y] in two variables, you can use the Hilbert's Nullstellensatz theorem. This theorem states that for any ideal I in the polynomial ring, the variety V(I) (set of points satisfying all polynomials in I) is equal to the radical of I, where the radical of an ideal is the set of all elements that when raised to a positive power are contained in the ideal. Therefore, the prime ideals of the polynomial ring are the radical ideals.

3. Can you give an example of a prime ideal in the polynomial ring C[x, y]?

One example of a prime ideal in the polynomial ring C[x, y] is the ideal (x, y), which consists of all polynomials with no constant term. This is a prime ideal because it is a radical ideal (the radical of (x, y) is also (x, y)) and it is closed under multiplication.

4. How many prime ideals are there in the polynomial ring C[x, y]?

There are infinitely many prime ideals in the polynomial ring C[x, y]. This is because the set of all polynomials with no constant term (such as the ideal (x, y) mentioned in the previous question) is just one example of a prime ideal, and there are infinitely many other possible combinations of polynomials that could form prime ideals.

5. How are prime ideals related to irreducible polynomials?

In the polynomial ring C[x, y], every prime ideal is generated by one irreducible polynomial. This means that if a polynomial is irreducible (cannot be factored into smaller polynomials), then it will generate a prime ideal. Additionally, if a polynomial is not irreducible, then it will generate a proper ideal (not a prime ideal).

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