Exercise 16, § 3

Let B be a discrete valuation ring whose residue field $k$ is finite and is not a prime field; let $k_0$ be the prime subfield of $k$ and let A be the subring of B consisting of the elements whose classes in the residue field belong to $k_0$. Let $\pi$ be a uniformizer of B and $(\theta_i)_{1 \leq i \leq m}$ a system of invertible elements of B such that the classes $\bar{\theta}_i \mod \pi$ of the $\theta_i$ form a system of representatives of $k^* \mod k_0^*$. Show that the elements $p_i = \theta_i \pi$ and the element $\pi$ are extremal in A and that every element in A is a product of an invertible element and powers of the $p_i$ and $\pi$, although A is not integrally closed.