Exercise 25, § 7

Let $k$ be a field, $A$ a local $k$-algebra, localized at a prime ideal of a finitely generated $k$-algebra. Prove that $A$ is regular if and only if $A$ is integral and e(A) = 1. (One may draw inspiration from the method followed in Exercise 24 or else remark that since $A$ is an integral quotient of a regular ring, the local ring Ā is equidimensional and has no immersed prime ideals.)