TITLE: Why is $Q[\pi]$ not a field?
QUESTION [10 upvotes]: I am having trouble seeing how to apply the definition of transcendental to see this. Thanks!

REPLY [11 votes]: Hint $\ $ Notice $\:\pi\:$ transcendental over $\rm\Bbb Q\:\Rightarrow\:\Bbb Q[\pi]\cong \Bbb Q[x].\:$ But a polynomial ring cannot be a field since if $\rm\ x^{-1}\! = f(x)\in\Bbb Q[x]\ $ then $\rm\ x \; f(x) = 1 \: \Rightarrow\:  0 = 1,\ $ by evaluating at $\rm\ x = 0. $
Remark $\ $ The above proof has a very instructive universal interpretation.