I've come across this problem in Etingof's notes on representation theory (Problem 5.1 on p. 78). It just sounds nice exercise... The question is :
Let $f : k(x_1,\ldots,x_n)\rightarrow k(y_1,\ldots, y_m)$ be a field embedding of field of rational functions. Show that $m\geq n$.
P.S: His hint was first to show for $f : k[x_1,\ldots,x_n]\rightarrow k(y_1,\ldots, y_m)$