Example 2

Let A be an integral, separated and complete local ring. Suppose that the residue field $k$ of A is finite, with $q = p^f$ elements, hence perfect of characteristic $p$. We have $x^q = x$ for every $x \in k$, whence $s^q = s$ for every multiplicative representative s. It follows that the set of multiplicative representatives consists of 0 and the $q - 1$ $(q - 1)$-th roots of unity in the field of fractions of A. If the field of fractions of A is locally compact, the existence of multiplicative representatives also follows from VI, § 9, No. 2, Proposition 3 (cf. also VI, § 9, Exercise 5).