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 ?



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
> 
> 
> 1. $p$ is surjective and both open and closed (i.e., maps open/closed
> sets to open/closed sets).
> 2. 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.


