The following result deals with the case of finite type affine schemes over an arbitrary field $k$.

Theorem: Let $A$ be a finitely generated algebra over a field $k$. Let $\iota: A \rightarrow \overline{A} = A \otimes_k \overline{k}$.
a) For every maximal ideal $\mathfrak{m}$ of $A$, the set $\mathcal{M}(\mathfrak{m})$ of maximal ideals $\mathcal{M}$ of $\overline{A}$ lying over $\mathfrak{m}$ is finite and nonempty.
b) The natural action of $G = \operatorname{Aut}(\overline{k}/k)$ on $\mathcal{M}(\mathfrak{m})$ is transitive. Thus $\operatorname{MaxSpec}(A) = G \backslash \operatorname{MaxSpec}(\overline{A})$.
c) If $k$ is perfect, the size of the $G$-orbit on $\mathfrak{m} \in \operatorname{MaxSpec}(A)$ is equal to the degree of the field extension of $k$ generated by the coordinates in $\overline{k}^n$ of any $\mathcal{M}$ lying over $\mathfrak{m}$.

In brief, the closed points correspond to the Galois orbits of the geometric points.

This is Theorem 8 in http://www.math.uga.edu/~pete/8320notes3.pdf.

The proof is left as an exercise, with some suggestions.

Exactly where this result came from, I cannot now remember. The text for the course that these notes accompany was Qing Liu's Algebraic Geometry and Arithmetic Curves (+1!), so it's a good shot that there is at least some cognate result in there.

The following result deals with the case of finite type affine schemes over an arbitrary field $k$.

Theorem: Let $A$ be a finitely generated algebra over a field $k$. Let $\iota: A \rightarrow \overline{A} = A \otimes_k \overline{k}$.
a) For every maximal ideal $\mathfrak{m}$ of $A$, the set $\mathcal{M}(\mathfrak{m})$ of maximal ideals $\mathcal{M}$ of $\overline{A}$ lying over $\mathfrak{m}$ is finite and nonempty.
b) The natural action of $G = \operatorname{Aut}(\overline{k}/k)$ on $\mathcal{M}(\mathfrak{m})$ is transitive. Thus $\operatorname{MaxSpec}(A) = G \backslash \operatorname{MaxSpec}(\overline{A})$.
c) If $k$ is perfect, the size of the $G$-orbit on $\mathfrak{m} \in \operatorname{MaxSpec}(A)$ is equal to the degree of the field extension of $k$ generated by the coordinates in $\overline{k}^n$ of any $\mathcal{M}$ lying over $\mathfrak{m}$.

In brief, the closed points correspond to the Galois orbits of the geometric points.

This is Theorem 8 in http://alpha.math.uga.edu/~pete/8320notes3.pdf.

The proof is left as an exercise, with some suggestions.

Exactly where this result came from, I cannot now remember. The text for the course that these notes accompany was Qing Liu's Algebraic Geometry and Arithmetic Curves (+1!), so it's a good shot that there is at least some cognate result in there.

The following result deals with the case of finite type affine schemes over an arbitrary field $k$.

Theorem: Let $A$ be a finitely generated algebra over a field $k$. Let $\iota: A \rightarrow \overline{A} = A \otimes_k \overline{k}$.
a) For every maximal ideal $\mathfrak{m}$ of $A$, the set $\mathcal{M}(\mathfrak{m})$ of maximal ideals $\mathcal{M}$ of $\overline{A}$ lying over $\mathfrak{m}$ is finite and nonempty.
b) The natural action of $G = \operatorname{Aut}(\overline{k}/k)$ on $\mathcal{M}(\mathfrak{m})$ is transitive. Thus $\operatorname{MaxSpec}(A) = G \backslash \operatorname{MaxSpec}(\overline{A})$.
c) If $k$ is perfect, the size of the $G$-orbit on $\mathfrak{m} \in \operatorname{MaxSpec}(A)$ is equal to the degree of the field extension of $k$ generated by the coordinates in $\overline{k}^n$ of any $\mathcal{M}$ lying over $\mathfrak{m}$.

In brief, the closed points correspond to the Galois orbits of the geometric points.

This is Theorem 8 in http://www.math.uga.edu/~pete/8320notes3.pdf.

The proof is left as an exercise, with some suggestions.

Exactly where this result came from, I cannot now remember. The text for the course that these notes accompany was Qing Liu's Algebraic Geometry and Arithmetic Curves, so it's a good shot that there is at least some cognate result in there.

The following result deals with the case of finite type affine schemes over an arbitrary field $k$.

Theorem: Let $A$ be a finitely generated algebra over a field $k$. Let $\iota: A \rightarrow \overline{A} = A \otimes_k \overline{k}$.
a) For every maximal ideal $\mathfrak{m}$ of $A$, the set $\mathcal{M}(\mathfrak{m})$ of maximal ideals $\mathcal{M}$ of $\overline{A}$ lying over $\mathfrak{m}$ is finite and nonempty.
b) The natural action of $G = \operatorname{Aut}(\overline{k}/k)$ on $\mathcal{M}(\mathfrak{m})$ is transitive. Thus $\operatorname{MaxSpec}(A) = G \backslash \operatorname{MaxSpec}(\overline{A})$.
c) If $k$ is perfect, the size of the $G$-orbit on $\mathfrak{m} \in \operatorname{MaxSpec}(A)$ is equal to the degree of the field extension of $k$ generated by the coordinates in $\overline{k}^n$ of any $\mathcal{M}$ lying over $\mathfrak{m}$.

In brief, the closed points correspond to the Galois orbits of the geometric points.

This is Theorem 8 in http://www.math.uga.edu/~pete/8320notes3.pdf.

The proof is left as an exercise, with some suggestions.

Exactly where this result came from, I cannot now remember. The text for the course that these notes accompany was Qing Liu's Algebraic Geometry and Arithmetic Curves (+1!), so it's a good shot that there is at least some cognate result in there.