Exercise 16, § 2

Let $A$ be an integral domain, $K$ its field of fractions, $K'$ a finite extension of $K$ of degree $n$ and $A'$ a subring of $K'$ containing $A$, with $K'$ as field of fractions and integral over $A$. Let $m$ be a maximal ideal of $A$ and $B$ the local ring $A[X]_{mA[X]}$ (Exercise 15).

(a) The subring $B' = B[A']$ of $K'(X)$ is integral over $B$; its field of fractions is $K'(X)$ and $[K'(X):K(X)] = n$.

(b) For every maximal ideal $m'$ of $A'$ containing $m$, $n' = m'B'$ is a maximal ideal of $B'$; if we write $n = mB$, then

and (Observe that $(A'/m') \otimes_A B$ is isomorphic to the field $(A'/m')(X)$ and deduce that $B'/n'$ is isomorphic to $(A'/m')(X)$.)

(c) The mapping $m' \mapsto m'B'$ is the bijection of the set of maximal ideals of $A$ containing $m$ onto the set of maximal ideals of $B'$; the inverse bijection is $n' \mapsto n' \cap A'$.