Let $p$ be a prime and let $Q$ be the set of quadratic residues and $N$ the set of nonresidues. Assume $2 \in Q$. When I look a the cyclotomic cosets mod $p$, I get ${\{0}\}, {\{Q}\}, {\{N}\}.$ For example, for $p = 7,17,23$. Is this true in all cases?
What's the theory behind this?