论文标题
数字字段元素中的星座
Constellations in prime elements of number fields
论文作者
论文摘要
鉴于任何数字字段，我们证明存在由整数环的成对非缔合元素组成的任意形状的星座。该结果扩展了著名的绿色定理，内容涉及理性素数和陶定理的高斯素数定理。此外，我们证明了具有整数系数的二元二元形式的主要表示形式的星座定理。更确切地说，对于非二元二进制二元二级二次形式$ f $而不是负面的，存在任意形状的星座，由成对的整数$（x，y）$组成，$ f（x，y）$是合理的素数。后者定理是通过将框架从整数的环扩展到一对秩序及其可逆的分数理想来获得的。
Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.
