Cyclotomic scheme is a Association scheme

I try to show that the following defines an association scheme: Let $\mathbb{F}_q$ be a field, $\omega$ a primitive element of $\mathbb{F}_q^\times$ and $s$ divides $q-1$. Define $r=\frac{q-1}{s}$, $C_0=\{0\}$, $C_1=\langle \omega^s\rangle=\{\omega^{sk}:k=0,1,\ldots,r-1\}$ and$$C_i=\omega^{i-1}C_1\quad\text{ for }i=2,\ldots,s.$$I have to show that the $C_i$ partition the set $\mathbb{F}_q$. I tried to construct an group operations which has the $C_i$ as its orbits, but so far I did not succeed. Does anyone have an idea?

Sincerely, Hypertrooper

• See the answer below. – Hypertrooper Jun 11 '18 at 16:10

1 Answer

Sometimes, it is to difficult to see the easiest solution. The $C_i$ are the equivalence classes of the quotient $\mathbb{F}_q^*/C_1$.