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- Jun 22, 2012

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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

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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

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