I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore.
It is metioned that if $\mathbb{K}$ is a $p$-adic field in his thesis tate construct a continuous map $\epsilon: \mathbb{K}^+ \rightarrow S^1$. This is done to define the analogue for the expression $\exp(2\pi i)$ in the $p$-adic Lie group setting.
I do not have the access to Tate's thesis, but I would like to know how one can construct such map. If someone could give me some hint, I would apreciate.
It’s really easy enough to do this. Just to give an example, I’ll construct (using Axiom of Choice) a map from ${\mathbb Q_p}^+$ to $S^1$, with kernel equal to $\mathbb Z_p$. First, note that $\mathbb Q_p/\mathbb Z_p$ is the union (direct limit if you prefer) of cyclic groups of order $p^n$, all $n$. This is also true of the group of complex numbers that are $p^n$-th roots of unity for all $n$. Now you use AC to get an isomorphism between these two groups.