Characters of the multiplicative group of a finite field

The group \(\mathbb F_q^\times\) is cyclic of order \(q-1\). Choose a generator \(g\) and put \[ \zeta=e^{2\pi i/(q-1)}. \] For each \(k\in\mathbb Z/(q-1)\mathbb Z\), define \[ \chi_k(g^m)=\zeta^{km}. \] This is well defined and multiplicative, hence a character \[ \chi_k:\mathbb F_q^\times\to\mathbb C^\times. \] Conversely, any character is determined by its value on \(g\), and that value must satisfy \[ \chi(g)^{q-1}=\chi(g^{q-1})=1. \] Thus \(\chi(g)\) is one of the \(q-1\) roots \(\zeta^k\), so every character is exactly one \(\chi_k\).

Therefore \[ \widehat{\mathbb F_q^\times} =\operatorname{Hom}(\mathbb F_q^\times,\mathbb C^\times) \cong C_{q-1}. \] For odd \(q\), the unique character of order \(2\) is the quadratic character.