I am very new to field theory and I am trying to prove that if $F$ is a field and $R \in F(x_1,x_2,\ldots,x_n):=\{ PQ^{-1} : P,Q \in F[x_1,x_2,\ldots,x_n] \}$ is nonconstant, then $R$ is transcendental over $F$.
Suppose not. Then we choose a polynomial $A=a_0+a_1x+\ldots+a_kx^k$ such that $A(R)= 0$. I want to say something along the lines of $A(R)\in F(x_1,x_2,\ldots,x_n)$ and has finite degree, so it can't have too many roots. But then $A(R)=0$ for all $(x_1,\ldots,x_n)$ so it does have too many roots.
If $F$ were infinite I can imagine that this works, though I don't know how to formulate it. But I have no idea what to do if $F$ is finite, so perhaps there is just a better way of doing it?
Any help would be greatly appreciated.