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 Abstract Algebra Abstract Algebra Math Forum

 August 2nd, 2017, 05:30 AM #1 Member   Joined: Oct 2013 Posts: 36 Thanks: 0 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? August 2nd, 2017, 05:50 AM #2 Senior Member   Joined: Aug 2017 From: United Kingdom Posts: 311 Thanks: 109 Math Focus: Number Theory, Algebraic Geometry Looks good to me Tags commutative algebra, empty, ideal, variety Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post joshis1 Algebra 2 October 10th, 2014 03:55 PM barokas Applied Math 4 September 25th, 2013 03:47 PM guru123 Algebra 4 January 13th, 2013 09:05 AM outsos Applied Math 36 April 30th, 2010 10:46 AM

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