How to prove that embedding (here F is an arbitrary algebraically closed field) $F[x_1$, $\ldots$, $x_n]$ into $F(x_1$, $\ldots$, $x_m)$ exists if and only if $n \leq m$?
The hint in the book I'm reading now was to calculate how grows the dimension of $W_n$ (here $W_n$ denotes the subspace of all polynomials with degree $\leq n$) and the dimension of its image under embedding. But I haven't any ideas about how can it help...
Can someone help me?
I'd use different indeterminates, for better clarity. Suppose you have an embedding $$ \varphi\colon F[x_1,\dots,x_n]\to F(y_1,\dots,y_m) $$ Then this can be uniquely extended to an embedding $$ \hat{\varphi}\colon F(x_1,\dots,x_n)\to F(y_1,\dots,y_m) $$ and the image has transcendence degree $n$ over $F$. Since $F(y_1,\dots,y_m)$ has transcendence degree $m$, it follows that $n\le m$.
See a paper by Yifan Wu for proofs (or a good algebra book).