Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$.
I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the roots of unit in $A$ to the roots of unit in this field as a bijection. It is easy to show that the projection of a root of unit is a root of unit in $F$, and Hensel's lemma allow us to conclude that this is surjective. The problem is, I cannot figure how to prove it is injective, ie, if a root of unit, $x$, in $A$ is congruent to $1$ mod $m$, where $m$ is the only maximal ideal of $A$, then $x$ must be $1$. (Because then the kernel of the multiplicative homomorphism we have defined is trivial).
Does someone have an idea?