I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first:
Es sei eine Anzahl $m$ von ganzen rationalen Funktionen $X_1,\dots,X_m$ der $n$ Variablen $x_1,\dots,x_n$ vorgelegt: \begin{equation}\begin{split}X_1&=f_1(x_1,\dots,x_n)\\&\vdots\\X_m&=f_m(x_1,\dots,x_n).\end{split}\end{equation} (He calls this system of substitutions (S)). Jede ganze rationale Verbindung von $X_1,\dots,X_m$ wird offenbar durch Eintragung dieser Ausdrücke notwendig stets eine ganze rationale Funktion von $x_1,\dots,x_n$. Es kann jedoch sehr wohl gebrochene rationale Funktionen von $X_1,\dots,X_m$ geben, die nach Ausführung jener Substitution (S) zu ganzen Funktionen in $x_1,\dots,x_n$ werden. Eine jede solche rationale Funktion von $X_1,\dots,X_m$, die nach Ausführung der Substitution (S) ganz in $x_1,\dots,x_n$ wird, möchte ich eine relativganze Funktion von $X_1,\dots,X_m$ nennen. Jede ganze Funktion von $X_1,\dots,X_m$ ist offenbar auch relativganz; ferner ist die Summe, die Differenz und das Produkt relativganzer Funktionen stets wiederum relativganz.
Das entstehende Problem ist nun: zu entscheiden, ob es stets möglich ist, ein endliches System von relativganzen Funktionen von $X_1,\dots,X_m$ aufzufinden, durch die sich jede andere relativganze Funktion in ganzer rationaler Weise zusammensetzen läßt.
Okay, unfortunately this is in German. Here's what I made out of it, but can't guarantee it is right:
Start with $f_1,\dots,f_m\in k[x_1,...,x_n]$ for some field $k$. For every $g\in k[X_1,\dots,X_m]$, $g(f_1,\dots,f_m)$ is then a polynomial in the $x_i$. But there can be $\varphi\in k(X_1,\dots,X_m)$ such that $\varphi(f_1,\dots,f_m)\in k[x_1,\dots,x_n]$, which Hilbert calls "relativganz". Every polynomial in the $X_i$ is of course already relativganz, as well as sum, difference, and product of functions which are relativganz.
The problem: Is it always possible to find finitely many r-g functions $g_1,\dots,g_s$ in $k(X_1,\dots,X_m)$ such that every r-g function $\varphi$ can be expressed in the $g_i$ in a polynomial way?
Now the formulation as it is mentioned on wikipedia:
Let $k$ be a field, and $K\subseteq k(x_1,\dots,x_n)$ be a subfield. Is the ring $R=K\cap k[x_1,\dots,x_n]$ finitely generated as a ring?
So, as for the original formulation, we can choose $R$ to be the ring of r-g functions. But what is $K$ in this case? I feel like the choice of $f_1,\dots,f_m$ should correspond one-to-one to the choice of the subfield $K$. Is it possible, given the second formulation, to translate this back to the notion of "relativganz", i.e., can I always interpret $R$ as a ring of r-g functions? Are they really equivalent?
Sorry for the many question in this last paragraph, but I want to really understand how to translate between those two seemingly different formulations :-)
Thank you very much in advance!