This is exercise 3.6.J from the most recent Vakil's notes.
Suppose that k is a field, and A is a finitely generated k-algebra. Show that closed points of Spec A are dense, by showing that if f ∈ A, and D(f) is a nonempty (distinguished) open subset of Spec A, then D(f) contains a closed point of Spec A. Hint: note that $A_f$ is also a finitely generated k-algebra. Use the Nullstellensatz 3.2.5 to recognize closed points of Spec of a finitely generated k-algebra B as those for which the residue field is a finite extension of k. Apply this to both B = A and B = $A_f$.
I know we have $A_f/pA_f=(A/p)_f$, then if p is a prime ideal of $A_f$, then the left side is a field and is a finite extension of k. Then how should I get the conclusion that $A/p$ is also a field?
Best Answer
We know that $(A/\mathfrak p)_f$ is finite-dimensional. Since $A/\mathfrak p$ is a domain and the image of $f$ in $A/\mathfrak p$ is nonzero, the map $A/\mathfrak p \to (A\mathfrak p)_f$ is injective. Hence $A/\mathfrak p$ is a domain that is a finite-dimensional $k$-algebra. Therefore, it is a field (exercise).