The Cassels heights of cyclotomic integers

TL;DR

This paper investigates the structure and properties of the set of mean square values of conjugates of cyclotomic integers, revealing its algebraic and order-theoretic characteristics, and analyzing subsets confined to specific rings.

Contribution

It characterizes the derived sets of the mean square values, computes their order type, and describes the structure of subsets within rings of cyclotomic integers, proving certain quadratic polynomials are universal.

Findings

Derived sets follow a linear pattern: ^{(k)}=(k+1).

The order type of  matches that of PV numbers.

Quadratic polynomials are proven to be universal.

Abstract

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.