Subextensions of Finitely Generated Fields Let $K$ be an extension of a field $F$, and assume that there exist $\alpha_1,\dots,\alpha_n \in K$ such that $K=F(\alpha_1,\dots,\alpha_n)$. Let $E$ be a subfield of $K$ containing $F$.
(I) Do there exist $\beta_1,\dots,\beta_m \in E$ such that $E=F(\beta_1,\dots,\beta_m)$?
(II) Assume that the previous question has (in general or in some specific example) a positive answer. Do there exist $\gamma_1,\dots,\gamma_k \in E$ such that $E=F(\gamma_1,\dots,\gamma_k)$ with $k \leq n$?
Any help is welcome. Thank you very much for your attention.
Comments. I have only a basic knowledge of field theory, which does not allow me to answer these general questions.
The only thing which is clear to me is that if $[K : F] < \infty$, then (I) has trivially a positive answer, since in this case $[E : F] < \infty$, so if $\beta_1,\dots,\beta_m$ are a basis of the $F$-vector space $E$, then clearly $E=F(\beta_1,\dots,\beta_m)$. As a corollary, if we assume that $K$ is an algebraic extension, then (I) has a positive answer, since being $K=F(\alpha_1,\dots,\alpha_n)$, we have in this case $[K : F] < \infty$. I do not know what can happen when $[K : F] = \infty$. As for (II), I have no idea at all of the answer.
 A: *

*For a finite extension $A/B$ then $A$ is finitely generated (as a field over $F$) iff $B$ is finitely generated. One direction is obvious. For the other one, if $A$ is finitely generated, then consider the extension $C$ generated (over $F$) by the coefficients of the $B$-minimal polynomials of the generators of $A$, then $A/C$ is a finite extension thus so is $B/C$, and hence $A$ finitely generated gives $C$ finitely generated thus $B$ finitely generated.


*Going back to the fields in your question, let $L=E(a_{k_1},\ldots,a_{k_s})$ where $a_{k_1},\ldots,a_{k_s}$ is a minimal subset of the $a_i$ such that $K/L$ is algebraic. $K$ is finitely generated and algebraic over $L$, thus $K/L$ is a finite extension, whence $L$ is finitely generated, by finitely many rational functions in $s$ variables (the $a_{k_j}$) and coefficients in $E$. Let $D$ be the extension of $F$ generated by the (finitely many) coefficients appearing in those rational functions. Then $D=E$ (use induction in $s$ to make it rigorous)


*For part II, I don't know. In characteristic $0$ we'll have that $E$ is a finite extension of a subfield of $K$ isomorphic to $F(t_1,\ldots,t_m)$ thus (by the primitive element theorem) $E\cong F(t_1,\ldots,t_m)[\alpha]$ is generated by $m+1\le n+1$ elements.
Is there an algorithm to check if a field like $\Bbb{Q}(x^3,y^3,x+y)$ is generated by 2 elements ?
