Hi:
If K/F is a finite field extension of degree n, so is K(x)/F(x), where K(x) is the field of rational functions in one variable over K, idem F(x).

I could prove K(x)/F(x) is a finite extension, but I cannot prove the degree is n. It helped me to have found that K(x) = K(p), where p is the polynomial p(x) = x. Analogously, F(x) = F(p). Any hint will be welcome. Thanks for reading.

is finite iff it's algebraic and finitely generated. Let be generators of the extension try showing these also generate over (I'm not really sure it works but seems a good place to start).

It's what I did. I proceeded like this: let K:F=n. Let a_1,...,a_n be a basis of K over F. Then K= F(a_1,...,a_n). I know that
K(x)= K(p) and F(x)= F(p), (1)
where p belonging to F[x] is given by p(x)= x. Substituting,
K(p)= F(a_1,...,a_n)(p)= F(p)(a_1,...a_n). (2)
Because K/F finite, K/F algebraic and because a_i belongs to K, a_i algebraic over F and, all the more so, algebraic over F(p). Hence, F(p)(a_1,...,a_n)/F(p) is a finite extension. Keeping in mind (1) and (2), K(x)/F(x) is finite.

This done, I tried to prove a_1,...,a_n generate K(x) over F(x). Or that they are linearly independent over F(x). But in vain. Anyways, thank you for your post.

Enrique.