Subextension of a finitely generated extension of fields If $E/K$ is a finitely generated field extension and $F$ is an intermediate field how can I prove that $F/K$ is finitely generated?
 A: The result is easy if you suppose that the big extension $E/K$ is algebraic ( equivalently: finite) and Benjamin has given you a proof.    
But it is also true in complete generality: if $E/K$ is a field extension and if $E=K(x_1,...,x_n)$ is a finitely generated extension, then any intermediate field $K\subset F\subset E$ is also finitely generated .  
The difficulty is that some or all of the $x_i$'s might be transcendental over $K$.
Even in the case of a purely transcendental extension $K\subset K(X_1,...,X_n)$ the situation is quite complicated and it is not true  for $n\gt 1$ that $F$ must be  purely transcendental too  : this is the failure of Lüroth's theorem in higher dimensions.  
A proof of finite generation of $F$  in the general non-algebraic case is surprisingly difficult to locate in the literature.  The only reference I could find is Theorem 24.9  in Isaacs's book.
A: A proof of the general case can be found in $\S 11.5$ of my field theory notes.
A: First of all if $E/K$ is finitely generated this means that $E = K(a_1,\ldots a_n)$ where the $a_i$ are algebraic over $K$. Since $F$ is an intermediate field, you have the containment
$$K \subseteq F \subseteq E.$$
We now need the result the following result: 


If $E = K(a_1,\ldots a_n)$, then $[E:K]$ finite.


$\textbf{Proof:}$ Since $a_1$ is algebraic over $F$, $[K(a_1) : K]$ is finite. Since $a_2$ is algebraic over $K$, it is algebraic over $K(a_1)$ because $K \subset K(a_1)$. Hence $[K(a_1,a_2):K]$ is finite by the dimension counting formula. Continuing in this fashion we see that $[E:K] = [K(a_1, \ldots a_n):K]$ is finite, proving our claim.
Since $[E:K]$ is finite, the dimension counting formula implies that $[E:F]$ and $[F:K]$ are finite. In particular this means that $[F:K]$ is finite so that $F/K$ is finitely generated.
Alternatively if you know linear algebra, $F$ being an intermediate field is a $K$ - vector subspace of a finite dimensional $K$ - vector space $E$, hence is finite dimensional as well.
