Algebraic Geometry - D&F Section 15.1, Exercise 19

Peter

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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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