TITLE: Generalization of Weak Nullstellensatz?
QUESTION [9 upvotes]: I believe the following is standard, namely when $k = \bar{k}$ is algebraically closed there is a bijection between points and maximal ideals
\begin{eqnarray*}
k^n &\longrightarrow& \operatorname{Specm}(k[X_1, \ldots, X_n]) \\
x &\longrightarrow& \ker(\operatorname{ev}_x)
\end{eqnarray*}
where surjectivity follows from Zariski's Lemma. It seems like the following should also be true, by essentially the same argument, but I couldn't find a reference. For $k$ not algebraically closed there is a bijection
\begin{eqnarray*}
\bar{k}^n / \operatorname{Aut}(\bar{k} / k) &\longrightarrow \operatorname{Specm}(k[X_1, \ldots, X_n])
\end{eqnarray*}
Is there a canonical reference for this ?

REPLY [3 votes]: 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.