Let $p$ be an odd prime. Assume $2$ is a quadratic residue modulo $p$. Is it true that the $(2)-$ cyclotomic cosets modulo $p$ are ${\{0}\}, {\{Q}\}, {\{N}\}$, where $Q$ are the quadratic residues modulo $p$ and $N$ are the non-residues?
My thoughts;
I looked at the $(2)$ - cyclotomic coset modulo $p$ containing $1$, which is always a quadratic residue. The coset then consists of successive powers of $2$ modulo $p$. I can't see why these are all the quadratic residues, though?
You're right, the maximal size of such a coset is roughly $\log_2 p$ rounded up but the number of quadratic residues is roughly $p/2.$ Now take $p$ large enough.