Algebraic Geometry - Affine Algebraic Sets

[Math Amateur]

Dummit and Foote, Exercise 20, Section 15.1 reads as follows:

If f and g are irreducible polynomials in \(\displaystyle k[x,y] \) that are not associates (do not divide each other), show that \(\displaystyle \mathcal{Z} (f,g) \) is either the empty set or a finite set in \(\displaystyle \mathbb{A}^2 \).

I am somewhat overwhelmed by this problem and do not get much insight from D&F's hints on this exercise.

I would therefore appreciate someone heling me with an approach to this problem.

Peter

 

[jvicens]

Hello Peter,

I understand your frustration with this problem. It can be quite challenging to approach at first glance. However, there are a few key concepts that can help guide your approach.

First, let's define some terms to make sure we are on the same page. An irreducible polynomial is one that cannot be factored into two non-constant polynomials. Associates in this context refer to polynomials that differ only by a unit factor, such as 2x and 4x. In other words, they are essentially the same polynomial.

Now, let's think about what the statement is asking us to prove. We have two polynomials, f and g, that are not associates. This means they cannot divide each other. We are looking at the set of points in the plane (represented by \mathbb{A}^2) where both f and g are equal to 0, also known as the zero set. The question is asking us to show that this set is either empty or finite.

To approach this problem, we can consider the contrapositive statement. This means we can assume that the zero set is infinite and try to derive a contradiction. We know that if the zero set is infinite, there must be an infinite number of points where both f and g are equal to 0. This means that there must be an infinite number of solutions to the system of equations f(x,y) = 0 and g(x,y) = 0.

Now, let's think about what this means for the polynomials f and g. If there are an infinite number of solutions to this system of equations, it means that f and g must have a common factor. This is because every solution to the system of equations must satisfy both f(x,y) = 0 and g(x,y) = 0, which means the solution must satisfy their common factor as well.

However, we know that f and g are not associates, so they cannot have a common factor. This leads to a contradiction, which means our assumption that the zero set is infinite must be false. Therefore, the zero set must be either empty or finite.

I hope this helps guide your approach to this problem. Let me know if you have any further questions or need clarification on any of the concepts. Good luck!