关于次数为2q的多项式的无平方因子值
On the Squarefree Values of Degree-$2q$ Polynomials
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中文总结 AI 辅助
本研究证明满足特定伽罗瓦条件的2q次首一不可约整系数多项式有无穷多整数n使其值为无平方因子数，并据此构造了单位形状收敛到六边形格的非伽罗瓦全实三次域无穷子族。
中文摘要 AI 辅助
在本研究中，我们证明：若h(x)是次数为2q（q为素数）的首一不可约整系数多项式，其定义的ℚ的域扩张包含次数为q的伽罗瓦扩张，则存在正整数密度的整数n使得h(n)为无平方因子数；特别地，有无穷多整数n使得h(n)为无平方因子数。作为应用，我们证明例外三次域族包含一个无穷子族，其单位形状收敛到六边形格。据我们所知，这是首个非伽罗瓦全实三次域族的单位形状收敛到六边形格的例子。
英文摘要
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.
