Context: I've just started reading Tate's thesis. In it, we start with a local field $k.$ The aim of the section is to describe the structure of the character groups of $k^+$ (the additive group) and $k^\*$(the multiplicative group). But for some reason when looking at the character group for $k^+$, we are looking only for the characters $\chi: k^{+} \to S^1$, where $S^1$ is the circle group but in $k^\*$, we are looking at quasi characters $\chi^\prime:k^\* \to \mathbb{C}^\*$. Why are we doing this? @anon [answered](https://math.stackexchange.com/a/868840) a related question, [Characters of a Group: two definitions](https://math.stackexchange.com/questions/868835/characters-of-a-group-two-definitions), on Math StackExchange regarding this but it really doesn't help much.



Why are we defining character groups differently for additive and multiplicative group in Tate's thesis?
I think the basic reason for the apparently different definitions boils down to the different topologies of $k^{\*}$ versus $k^{+}$.


For simplicity of discussion, let's consider the case that $k= \mathbb{Q}\_p$. If $\psi: \mathbb{Q}\_p^{+}$ is a continuous additive character, then it must have image in the unit circle. A simple way to see this is to use that $\mathbb{Z}\_p$ is compact, so its image under the continuous map $\psi$ is a compact subgroup of $\mathbb{C}^{\*}$. It therefore lies in $S^1$. One can apply the same argument to the compact subgroups $p^{-1} \mathbb{Z}\_p$, $p^{-2} \mathbb{Z}\_p$, and so on.


On the other hand, one can easily define continuous quasi-characters of $\mathbb{Q}\_p^\*$ that do not have image in the unit circle. For instance, take $\chi(p) = p$ and $\chi(x) = 1$ for $x \in \mathbb{Z}\_p^{\*}$. It is still true that $\mathbb{Z}\_p^\*$ is a compact subgroup of $\mathbb{Q}\_p^\*$, but now we have $\mathbb{Q}\_p^\* = \bigcup\_{j \in \mathbb{Z}} p^j \mathbb{Z}\_p^{\*}$, and the character is free to send $p$ anywhere in $\mathbb{C}^\*$.


