分圆整数的Cassels高度
The Cassels heights of cyclotomic integers
摘要：
研究了分圆整数$β$共轭模的均方值集$\mathscr C$.对于它的$k$次导出集$\mathscr C^{(k)}$，我们证明了$\mathscr C^{(k)}=(k+1)\mathscr C\,\, (k\ge 0)$，因此也有${\mathscr C}^{(k)}+{\mathscr C}^{(\ell)}={\mathscr C}^{(k+\ell+1)}\,\,(k,\ell\ge 0)$。我们还计算了$\mathscr C$的序类型，并证明了它与PV数集的序类型相同。进一步，我们精确地描述了限制集$\mathscr C_p$，其中$β$局限于环$\mathbb Z[ω_p]$，其中$p$是奇素数，$ω_p$是本原$p$的单位根.为了做到这一点，我们证明了二次多项式$a^2+ab+b^2+c^2+a+b+c$和$a^2+b^2+c^2+ab+bc+ca+a+b+c$是通用的。
英文摘要：
We study the set $\mathscr C$ of mean square values of the moduli of the conjugates of cyclotomic integers $β$. 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 $β$ are confined to the ring $\mathbb Z[ω_p]$, where $p$ is an odd prime and $ω_p$ 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.
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