As a generalization of the finiteness result Hilbert proved in his 1890 paper, one usually formulates the following nowadays:
Let $G\to\operatorname{GL}(V)$ be a rational representation of a linearly reductive group $G$ on a finite-dimensional $k$-vector space $V$. Then $k[V]^G$ is a finitely generated $k$-algebra.
The sources I looked at so far assume either $k=\mathbb{C}$, $k$ algebraically closed, or $k$ infinite. In what generality does this result hold, regarding $k$?
From the non-constructive proof of the result, I can't really see any reliance on $k$ being algebraically closed; just the existence of a Reynolds operator is needed. But $k$ algebraically closed or infinite is often assumed when reductive algebraic groups are defined. Any reference for this result with $k$ arbitrary or whatever the maximum generality is is greatly appreciated!
Edit: If I remember correctly, Emmy Noether proved the case $k$ arbitrary and $G$ a finite subgroup of $\operatorname{GL}_n(k)$ acting on the polynomial ring $k[x_1,\dots,x_n]$. The only "problem" there could be is that in general $k[V]$ is not a polynomial ring if $k$ is a finite field.
In Ideals, Varieties, and Algorithms, by Cox, Little, and O'Shea, the theorem is shown for finite groups over any characteristic 0 field. The requirement that the field has characteristic 0 is clearly necessary if we want the Reynold's operator to be defined for arbitrary groups. The book doesn't really cover enough algebra to go into full generality for the linearly reductive case, so I'm not sure how this result carries over to that.