= Cyclotomic class of a finite field
{title2=$C_j=\gamma^j\langle\gamma^m\rangle,\quad m\mid q-1$}
= Cyclotomic classes of a finite field
{synonym}
The <cosets> of the index-$m$ <subgroup> of the <multiplicative group of a finite field> are its cyclotomic classes of index $m$. For a <character> with $\chi(\gamma)=\zeta_m$, their indicators are $\mathbf1_{C_j}(x)=m^{-1}\sum_{h=0}^{m-1}\zeta_m^{-jh}\chi^h(x)$, with every <character> extended by zero at zero. <Jacobi sums of finite-field characters> imply $|C_i\cap(t+C_j)|=q/m^2+O_m(\sqrt q)$ uniformly for $t\ne0$.
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