Stark Units

代数数论研究通过将一个代数数邻接到有理数域上而得到的域。斯塔克猜想假定这样一个数域的解析定义的L级数的特殊值与数域的代数性质之间存在着深刻的联系。特别是，这些猜想预言，从L级数得到的复数的某些特定元素实际上是代数元素，即斯塔克单位。在数域的相对阿贝尔扩张的情况下，猜想进一步预测，这些元素将由分析数据唯一地定义，直到选择单位根，生成数域的阿贝尔扩张。斯塔克猜想是代数论的核心问题之一。这项调查涉及检查与这些猜想有关的一些问题，继续和扩大调查人员先前的工作。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

Algebraic number theory concern the study of the fields obtained by adjoining an algebraic number to the field of rational numbers. The Stark Conjectures postulate a deep connection between special values of the analytically defined L-series of such a number field and algebraic properties of the field. In particular, these Conjectures predict that certain specific elements of the complex numbers obtained from the L-series are actually algebraic elements, the Stark units. In the case of a relative abelian extension of number fields, the Conjectures further predict that these elements, which would be uniquely defined by the analytic data up to a choice of a root of unity, generate abelian extensions of the number field. The Stark Conjectures are among the central questions in algebraic number theory. This investigation is concerned with examining a number of questions regarding these conjectures, continuing and extending prior work of the investigators. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.