Closed vs Rational Points on Schemes

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.

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