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 a related question, Characters of a Group: two definitions, on Math StackExchange regarding this but it really doesn't help much.

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 a related question, Characters of a Group: two definitions, on Math StackExchange regarding this but it really doesn't help much.

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? There is one answer on math stackexchange regarding this but it really doesn't help much. I'll link it here.

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 a related question, Characters of a Group: two definitions, on Math StackExchange regarding this but it really doesn't help much.