AI 中文总结
本文针对$\text{Q}^{(d)}$的整数环合成环元素的高度下界展开研究，证明$\text{Q}^{(3)}$对应的合成环具有Northcott性质，并引入子域大小概念，给出相关子域的大小与Northcott性质的结果。
AI 中文摘要
设$\boldsymbol{\text{Q}^{(d)}}$为所有次数不超过$d$的数域的合成域。2001年，Bombieri和Zannier证明了$\text{Q}^{(2)}$具有Northcott性质，并提出当$d\boldsymbol{\text{≥}}3$时会有什么结果的问题。本文研究这类数域的整数环合成环中元素的绝对Weil高度，特别地，将$\text{Q}^{(3)}$视为$\text{Q}^{(2)}$与极小无限三次域族的合成域，证明这些域的整数环合成环确实具有Northcott性质，该结果源于用次数表示的新高度下界。此外，本文为$\text{Q}^{(3)}$的子域引入了大小概念，例如$\text{Q}^{(3)}$的大小为1，其极大阿贝尔子域的大小为1/2，还证明存在一个大小为1的$\text{Q}^{(3)}$子域具有Northcott性质。
英文摘要
Let $\mathbb{Q}^{(d)}$ be the composite field of all number fields of degree at most $d$. In 2001 Bombieri and Zannier proved that $\mathbb{Q}^{(2)}$ has the Northcott property and asked what happens for $d\geq 3$. Here we study the absolute Weil height for elements in the composite ring of the rings of integers of such number fields. In particular, we consider $\mathbb{Q}^{(3)}$ as the composite field of $\mathbb{Q}^{(2)}$ and a minimal infinite family of cubic fields, and we show that the composite ring of the rings of integers of these fields does have the Northcott property. Our results follow from new height lower bounds, expressed in terms of the degree. Moreover, we introduce a notion of size for subfields of $\mathbb{Q}^{(3)}$. For instance, $\mathbb{Q}^{(3)}$ has size $1$ and the maximal abelian subfield of $\mathbb{Q}^{(3)}$ has size $1/2$. We show that there is a subfield of $\mathbb{Q}^{(3)}$ of size $1$ which has the Northcott property.
Comments10 pages
