标题： 某些具有Northcott性质的有理数的无限扩张 显示英文标题

标题： On certain infinite extensions of the rationals with Northcott property

摘要： 如果一个代数数的集合的每个有界Weil高度子集都是有限的，那么这个集合具有Northcott性质。 Northcott定理有许多丢番图应用，它指出有界次数的集合具有Northcott性质。 Bombieri、Dvornicich和Zannier提出了寻找具有此性质的无限次数域的问题。 Bombieri和Zannier证明了$\IQ_{ab}^{(d)}$，即由所有次数最多为$d$的代数数生成的域的最大阿贝尔子域，就是这样一个域。 在本文中，我们给出了Northcott性质的简单判据，并作为应用，我们推导出几个新的例子，例如 $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$具有Northcott性质当且仅当$2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$趋向于无穷大。 显示英文摘要

摘要： A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that $\IQ_{ab}^{(d)}$, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$ has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity.