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 class in the residue field belongs 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. $k_0^*$. Now prove that the elements $p_i = \theta_i \pi$ and the element $\pi$ are extremal in A and that every element of A is the product of an invertible element and powers of the $p_i$ and of $\pi$, though A is not integrally closed.