Generalization of Weak Nullstellensatz?
The closest reference in literature I have encountered is in Mumford's Red book of varieties and schemes, II.4 Theorem 1. More precisely, a direct citation is as follows:
Let $X_0$ be a prescheme over $k_0$, let $X = X_0 \times_{k_0} k$, and let $p: X \rightarrow X_0$ be the projection. Assume that $k$ is an algebraic closure of $k_0$. Then
- $p$ is surjective and both open and closed (i.e., maps open/closed sets to open/closed sets).
- For all $x,y \in X$, $p(x) = p(y)$ if and only if $x = \sigma_X(y)$, some $\sigma \in Gal(k/k_0 )$. In other words, for all $x \in X_0$ $p^{-1}(x)$ is an orbit of $Gal(k/k_0)$. Moreover, $p^{-1}(x)$ is a finite set.
The theorem applied to $X_0=\mathrm{Spec}\,k_0[x_1, \dots, x_n]$ together with the standard weak Nullstellensatz for $X=\mathrm{Spec}\,k[x_1, \dots, x_n]$ should give your statement.
Just discovered, it's demonstrated in Bourbaki Commutative Algebra Chapter V Section 3.3 Proposition 2.
Or see Proposition 2.4.6 in Bjorn Poonen's book Rational Points on Varieties (link). This is almost exactly the result you conjectured, just a bit more general:
Let $X$ be a $k$-variety. Then the map $$\left\{\text{$\operatorname{Gal}_k$-orbits in $X(\overline{k})$}\right\}\rightarrow \left\{ \text{closed points of $X$} \right\}$$ given by mapping the orbit of $f \colon \operatorname{Spec} \overline{k} \to X$ to $f(\operatorname{Spec} \overline{k} )$ is a bijection.