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.