The Cassels heights of cyclotomic integers

Abstract

We study the set $\mathcal{C}$ of mean square values of the moduli of the conjugates of cyclotomic integers $\beta$. For its $k$th derived set ${\mathcal{C}}^{(k)}$, we show that ${\mathcal{C}}^{(k)}=(k+1)\mathcal{C}(k\ge 0)$, so that also ${\mathcal{C}}^{(k)}+{\mathcal{C}}^{(\ell )}={\mathcal{C}}^{(k+\ell +1)}(k,\ell \ge 0)$. We also calculate the order type of $\mathcal{C}$, and show that it is the same as that of the set of PV numbers. Furthermore, we describe precisely the restricted set ${\mathcal{C}}_p$ 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.

13 pages