We study the set $\mathscr C$ of mean square values of the moduli of the conjugates of cyclotomic integers $\beta$. For its $k$th derived set $\mathscr C^{ (k)}$, we show that $\mathscr C^{ (k)}=(k+1)\mathscr C\,\, (k\ge 0)$, so that also ${ \mathscr C}^{ (k)}+{ \mathscr C}^{ (\ell)}={ \mathscr C}^{ (k+\ell+1)}\,\,(k,\ell\ge 0)$. We also calculate the order type of $\mathscr C$, and show that it is the same as that of the set of PV numbers. Furthermore, we describe precisely the restricted set $\mathscr C_p$ where the $\beta$ are confined to the ring $\mathbb Z[\omega_p]$, where $p$ is an odd prime and $\omega_p$ is a primitive $p$th root of unity. In order to do this, we prove that both of the quadratic polynomials $a^2+ab+b^2+c^2+a+b+c$ and $a^2+b^2+c^2+ab+bc+ca+a+b+c$ are universal.
Categories: math.NT