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August 2nd, 2017, 05:30 AM   #1
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Ideal of the empty Variety

So in Atiyah-Macdonald in exercise 5.17 they ask you to show if $I(X) \neq (1)$ then $X$ is non-empty where $X$ is an affine algebraic variety in $k^n$, $k$ is an algebraically closed field and $I(X)$ denotes $\{f \in k[t_1,...,t_n] : f(x) = 0 \forall x \in X\}$.

They then go on to provide a hint using the previous (very complicated exercise) and all of the solutions I've seen seem to use the hint.

However why can't you just say that if $X$ is empty then vacuously $I(X) =k[t_1,...,t_n]$ which gives the result?
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August 2nd, 2017, 05:50 AM   #2
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Looks good to me
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