Exercise 3, § 9

Let K be a totally disconnected non-discrete locally compact commutative field; let A be the ring of the absolute value $\mathrm{mod}_K$ on K and let U be the group of units of A. For every integer $n > 0$, let $_nU$ denote the group of n-th roots of unity in K and $U^n$ the subgroup of U consisting of the n-th powers of the elements of U. Show that, if $n$ is prime to the characteristic of K, $U^n$ is an open subgroup of U and that

(use Exercise 14 of Integration, Chapter VII, § 2 and Commutative Algebra, Chapter III, § 4, no. 6, Corollary 1 to Theorem 2 to show that, if m is the maximal ideal of $\mathbf{A}$, the image under $x \mapsto x^n$ of $1 + m^k$ is $1 + n \cdot m^k$ for $k$ sufficiently large).