Cyclotomic cosets $\pmod 8$, $p=3$

Question

First time dealing with cyclotomic cosets, and I've solved my first exercise; I wish to have any kind of corrections/hints you can write to me.

Find all the cyclotomic cosets $\pmod 8$, $p=3$

Doing some calculation, considering that for a value $s$ here I have $\{s,3s\}$ because in my case $p^m-1=8, m=3$ we have:

$C_0=\{0\}$

$C_1=\{1,3\}$

$C_2=\{2,6\}$

$C_4=\{4\}$

$C_5=\{5,\color{red}{7}\}$

Is this right? I checked for every $s$ from $0$ to $8$. Is this solution ok?

Edit: added 7 to $C_5$ after some math.

Answer 1

I think you’re looking for the $(3)-$ cyclotomic cosets modulo $8 = 3^2 -1$. If this is the case, then your solution is nearly correct, but, note that the union of these cosets is precisely the integers from $0$ to $7$.

You are missing the number $7$, so your solution is wrong. But note that $5 \times 3 = 15 \equiv 7$ modulo $8$. So, in fact, $C_5 = {\{5,7}\}$. The rest of your cosets are correct.