Why does the integral domain “being trapped between a finite field extension” implies that it is a field?
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The following is an exercise from Qing Liu's Algebraic Geometry and Arithmetic Curves . Exercise 1.2. Let $varphi : A to B$ be a homomorphism of finitely generated algebras over a field. Show that the image of a closed point under $operatorname{Spec} varphi$ is a closed point. The following is the solution from Cihan Bahran. http://www-users.math.umn.edu/~bahra004/alg-geo/liu-soln.pdf. Write $k$ for the underlying field. Let’s parse the statement. A closed point in $operatorname{Spec} B$ means a maximal ideal $n$ of $B$ . And $operatorname{Spec}(varphi)(n) = varphi^{−1}(n)$ . So we want to show that $p := varphi{−1}(n)$ is a maximal ideal in $A$ . First of all, $p$ is definitely a prime ideal of $A$ and $varphi$ descends to an injective $k$ -algebra homomorphism $ψ :