Exercise 6, § 7

Let $k$ be a field, $A$ a local, complete, reduced $k$-algebra of dimension 1 and $m$ its maximal ideal. Now prove that the minimum number of generators of an ideal of $A$ is at most equal to $e_m(A)$ (there exists $X \in A$ such that the injection of $k[[X]]$ into $A$ makes $A$ a free $k[[X]]$-module of rank $e_m(A)$).