I am currently working through Algebraic Curves by W. Fulton, and I am having a rough time understanding the section "Modules; Finiteness Conditions". I have muscled through Fulton exercise 1.41 and 1.43, but I am still not confident.
1.44:Show that $L=K(X)$ is a finitely generated field extension of $K$, but $L$ is not ring-finite over $K$
Hint: If $L$ were ring-finite over $K$, a common denominator of ring generators would be an element $b\in K[X]$ such that for all $z\in L$, $b^nz\in K[X]$ for some $n$; but let $z=1/c$, where c doesn't divide $b$.