Exercise 27, § 1

Let $A$ be an integrally closed but not completely integrally closed domain and $K$ its field of fractions; let $a \neq 0$ be a non-invertible element of $A$ for which there exists $d \neq 0$ in $A$ such that $da^{-n} \in A$ for every integer $n \geqslant 0$.

Show that in the field of formal power series $K((X))$ there exists a formal power series $\sum_{k=1}^{\infty} u_k X^k = f(X)$ satisfying the equation

hence integral over $A[[X]]$, belonging to the field of fractions of $A[[X]]$ but not belonging to $A[[X]]$, so that $A[[X]]$ is not integrally closed.