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.