Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
[Lüroth theorem](https://en.wikipedia.org/wiki/L%C3%BCroth%27s_theorem) says that a field $L$, $k \subset L \subset k(x)$ containing a nonconstant polynomial over $k$ is equal to $k(h)$ for some $h \in k[x]$.
More precisely, in Lüroth theorem $h \in k(x)$, and E. Noether has proved that $h \in k[x]$ (see "Selected Topics on Polynomials" by A. Schinzel).
Unfortunately, I do not have the book of A. Schinzel (I have learnt about Noether's result from [Formanek's paper](https://www.researchgate.net/publication/266339075_Observations_about_the_Jacobian_conjecture)), so I do not have an available proof for this result.


Given $f=f(x),g=g(x) \in k[x]$, $k \subset k(f,g) \subseteq k(x)$,
Lüroth-Noether theorem says that $k(f,g)=k(h)$ for some $h=h(x) \in k[x]$. 



> 
> **(1)** What exactly can be said about $h$ in terms of $f$ and $g$? (Except, of course, that $h \in k(f,g)$).
> 
> 
> 


I have played a little with several $f$'s and $g$'s and I have not found $f$ and $g$ with $1<\gcd(\deg(f),\deg(g))<min\{\deg(f),\deg(g)\}$ such that $k(f,g)=k(x)$. For example: (i) $f=x^2,g=x^3$ with $\gcd(\deg(x^2),\deg(x^3))=1$. 
(ii) $f=x^3,g=x^6+x^2$ with $\gcd(\deg(x^3),\deg(x^6+x^2))=3=min\{\deg(f),\deg(g)\}$.



> 
> **(2)** Therefore, if $k(f,g)=k(x)$, then I conjecture that necessarily 
>  $\gcd(\deg(f),\deg(g))=1$ or $\deg(f) | \deg(g)$ or $\deg(g) | \deg(f)$:
>  Is my conjecture true or false? (Perhaps it is possible to find a counterexample to my conjecture with some help from algebraic geometry?).
> 
> 
> 


Any hints and comments are welcome! Thank you very much.



Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
This answer is just repeating comments, above, about a negative answer to conjecture (2). I'm unable to say anything for question (1).


The conjecture (2) is not correct. As you note, $k(x) = k(x^3,x^6+x^2)$; and then
$$
 k(x) = k(x^3 \cdot (x^6+x^2), x^6+x^2) = k(x^9+x^5, x^6+x^2),
$$
where $\deg(f)=9$ and $\deg(g)=6$, so $\gcd(\deg(f),\deg(g))=3>1$ but neither $\deg(f)$ nor $\deg(g)$ divides the other one. It is also possible to have counterexamples with nonzero constant coefficients. We have
$$
\begin{split}
 k(x) &= k(x+1) \\
&= k((x+1)^9+(x+1)^5, (x+1)^6+(x+1)^2) \\
&= k(x^9+\dotsb+2, x^6+\dotsb+2),
\end{split}
$$
so if $k$ has characteristic $\neq 2$ then $f$ and $g$ have nonzero constant coefficients. In characteristic $2$, we can find a counterexample with similar manipulations, such as
$$
 k(x) = k(x^3, x^9+x^6+x^2) = k(x^3 \cdot (x^9+x^6+x^2), x^9+x^6+x^2)
$$
followed by the same substitution of $x+1$ for $x$.


