Exercise 3, § 7

Let U be the set of invertible elements of A and $U' = 1 + m \subset U$. (a) Show that $U' \subset U_f$. (If $x = 1 + y$ with $y \in m$, then $x^{p^n}$ tends to 1 as n tends to $+\infty$ by the binomial theorem.) (b) Show that $U_f$ is the set of elements of U whose image in $A/m$ is a root of unity. (Use (a).) Hence recover the fact that $U = U_f$ when K is locally compact (A/m is then finite).