Exercise 16, § 2

Let A be a Priiferian domain, K its field of fractions and L a (finite or otherwise) algebraic extension of K; show that the integral closure $A'$ of A in L is a Priiferian domain. (Let $x \in L$ and let $a_0 X^n + a_1 X^{n-1} + \ldots + a_n$ be a polynomial in $A[X]$ of which $x$ is a root; if we write

show that the $b_i$ and the $b_i x$ belong to $A'$, using § 1, Exercise 12 (a). If $a = \sum_{i=0}^n A a_i, \ b = \sum_{j=0}^{n-1} B b_j$ show then that the ideal $b.Ba^{-1}$ is the inverse of $B + Bx$.)