论文标题
在完全复杂的四分之一的扩展中计算相对功率积分基础碱基
Calculating relative power integral bases in totally complex quartic extensions of totally real fields
论文作者
论文摘要
一段时间以前，我们将功率积分碱基发电机的单基因研究和计算扩展到了相对情况。到目前为止，我们考虑了复杂四分之一场的（通常是完全真实的）扩展。在本文中，我们考虑了完全真实领域的相对扩展中的功率积分基础。完全真实的数字字段的完全复杂的四分之一扩展似乎是我们在这里详细介绍的最简单情况。正如我们将看到的那样，即使在这种情况下，我们也必须克服一些意外的困难，但是我们可以通过正确（但不是琐碎）调整标准方法来解决这些困难。我们在一个明确的示例中演示了我们的一般算法。我们描述了如何在这种情况下修改四分之一相对扩展中的相对指标形式方程的一般方法。作为一种副产品，我们表明，完全真实领域的完全复杂扩展中的相对thue方程只能具有小的解决方案，并且我们构建了一种特殊的方法，用于枚举特殊单位方程的小解决方案。这些陈述也可以应用于其他毒液问题。
Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions of complex quartic fields. In the present paper we consider power integral bases in relative extensions of totally real fields. Totally complex quartic extensions of totally real number fields seems to be the most simple case, that we detail here. As we shall see, even in this case we have to overcome several unexpected difficulties, which we can, however solve by properly (but not trivially) adjusting standard methods. We demonstrate our general algorithm on an explicit example. We describe how the general methods for solving relative index form equations in quartic relative extensions are modified in this case. As a byproduct we show that relative Thue equations in totally complex extensions of totally real fields can only have small solutions, and we construct a special method for the enumeration of small solutions of special unit equations. These statements can be applied to other diophantine problems, as well.
