
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 2nd, 2017, 05:30 AM  #1 
Member Joined: Oct 2013 Posts: 36 Thanks: 0  Ideal of the empty Variety
So in AtiyahMacdonald in exercise 5.17 they ask you to show if $I(X) \neq (1)$ then $X$ is nonempty 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to master maths  should we solve simple problems or variety of problems?  joshis1  Algebra  2  October 10th, 2014 03:55 PM 
The empty set  barokas  Applied Math  4  September 25th, 2013 03:47 PM 
A variety of questions  guru123  Algebra  4  January 13th, 2013 09:05 AM 
empty set  outsos  Applied Math  36  April 30th, 2010 10:46 AM 