发表机构
University of Toyama（富山大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三次域族$K_b$，确定$\eta_b$为基本单位的正密度$b$集合，通过佩尔方程刻画其为单位平方的情形，并构造无限多个3-类域塔长度大于1的双二次域。
AI 中文摘要
对于整数$b\neq0,1$，设$\theta$为$f_b(x)=x^3-3bx-b^3$的唯一实根，并设$K_b=\mathbb{Q}(\theta)$。我们展示了一个正密度的$b$的显式集合，对于这些$b$，$\eta_b=-1/(\theta-(b+1))$是$K_b$的基本单位，并确定了使得$\eta_b$为单位平方的$b$的集合：该集合由佩尔方程$D^2-3E^2=1$参数化，因此是无限的，并且在关于域判别式的显式温和条件下，它涵盖了所有$\eta_b$不是基本单位的$b$。该条件仅对$|b|\le3000$中的四个$b$失效，且在其中之一$b=3$处，结论本身失效。相比之下，在由$\eta_b$生成的序$\mathbb{Z}[\eta_b]$中，$\eta_b$对每个$b$都是基本单位，因此这些例外是最大序的现象。作为应用，我们构造了无限多个双二次域，其$3$-类域塔长度大于$1$。
英文摘要
For an integer $b\neq0,1$ let $θ$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\mathbb{Q}(θ)$. We exhibit an explicit set of $b$ of positive density for which $η_b=-1/(θ-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $η_b$ is the square of a unit: it is parametrized by the Pell equation $D^2-3E^2=1$, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all $b$ for which $η_b$ is not the fundamental unit. That condition fails for only four $b$ with $|b|\le3000$, and at one of them, $b=3$, the conclusion itself fails. In the order $\mathbb{Z}[η_b]$ generated by $η_b$, by contrast, $η_b$ is the fundamental unit for every $b$, so these exceptions are a phenomenon of the maximal order. As an application we construct infinitely many biquadratic fields whose $3$-class field tower has length greater than $1$.
