Algebraic Geometry - D&F Section 15.1, Exercise 19

[Math Amateur]

Exercise 19 of Section 15.1 in Dummit and Foote reads as follows:

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19. For each non-constant \(\displaystyle f \in k[x] \) describe \(\displaystyle \mathcal{Z}(f) \subseteq \mathbb{A}^1 \) in terms of the unique factorization of \(\displaystyle f \) in \(\displaystyle k[x] \), and then use this to describe \(\displaystyle \mathcal{I}( \mathcal{Z} (f)) \). Deduce that \(\displaystyle \mathcal{I}( \mathcal{Z} (f)) = (f) \) if and only if \(\displaystyle f \) is the product of distinct linear factors in \(\displaystyle k[x] \).

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I can see that by the Fundamental Theorem of Algebra if \(\displaystyle k = \mathbb{C} \) that \(\displaystyle \mathcal{Z}(f) \) is a finite set of n or less points, and if \(\displaystyle k = \mathbb{R} \) or \(\displaystyle \mathbb{Q}\) then the possibilities for \(\displaystyle \mathcal{Z}(f) \) include the empty set. However I am not sure how to formally express these things and am not sure of the situation with finite fields. Further, I am not confident of a formal and rigorous solution to the rest of the problem.

I would appreciate some guidance.

Peter

 

[jvicens]

Dear Peter,

Thank you for your question. Let me provide some guidance on how to approach this exercise.

First, let's define the notation that is being used. The notation \mathcal{Z}(f) represents the set of points in \mathbb{A}^1 where the polynomial f vanishes, or in other words, the roots of the polynomial. This set can be described in terms of the unique factorization of f in k[x] . In particular, if f is a product of distinct linear factors in k[x] , then \mathcal{Z}(f) will consist of the roots of each of these factors, which will be distinct points in \mathbb{A}^1 .

Next, let's consider the ideal \mathcal{I}( \mathcal{Z} (f)) , which represents the set of polynomials that vanish on \mathcal{Z}(f) . In this case, since \mathcal{Z}(f) consists of distinct points, the ideal will consist of polynomials that are divisible by each of the linear factors in the unique factorization of f . In other words, the ideal will be generated by f itself, since f is a product of distinct linear factors.

Finally, we can deduce that \mathcal{I}( \mathcal{Z} (f)) = (f) if and only if f is the product of distinct linear factors. This is because if f is not the product of distinct linear factors, then there will be some repeated factors, which will result in a larger ideal that is not generated by f . On the other hand, if f is the product of distinct linear factors, then the ideal will be generated by f itself.

I hope this helps to clarify the exercise and provides a formal and rigorous solution. Please let me know if you have any further questions. Best of luck with your studies.