本书的目的是阐述代数数理论，不包括类场理论及其后果。发展这门学科有很多方法；最新的趋势是忽视经典的德金理想理论，转而采用局部方法。然而，对于将代数数应用于数论的其他领域所必需的数值计算，旧方法似乎更适合，尽管它的阐述显然更长。另一方面，如泰特的论文所示，局部方法在分析方面更为有效。
因此，作者试图调和这两种方法，在前四章中对经典观点进行了完整的阐述，然后转向局部方法。在第一章中，我们介绍了Dedekind域理论和估值理论的必要工具，包括Dedekind域上有限生成模的结构。在第2、3和4章中，发展了代数数的经典理论。第5章介绍了p-adic场理论的基本概念，第6章介绍了它们在代数数场研究中的应用。我们在这里介绍了沙法耶维奇对克罗内克-韦伯定理的证明，以及阿代尔斯和艾代尔斯的主要性质。
Elementary and Analytic Theory of Algebraic Numbers
The aim of this book is to present an exposition of the theory of alge braic numbers, excluding class-field theory and its consequences. There are many ways to develop this subject; the latest trend is to neglect the classical Dedekind theory of ideals in favour of local methods. However, for numeri cal computations, necessary for applications of algebraic numbers to other areas of number theory, the old approach seems more suitable, although its exposition is obviously longer. On the other hand the local approach is more powerful for analytical purposes, as demonstrated in Tate’s thesis.
Thus the author has tried to reconcile the two approaches, presenting a self-contained exposition of the classical standpoint in the first four chapters, and then turning to local methods. In the first chapter we present the necessary tools from the theory of Dedekind domains and valuation theory, including the structure of finitely generated modules over Dedekind domains. In Chapters 2, 3 and 4 the clas sical theory of algebraic numbers is developed. Chapter 5 contains the fun damental notions of the theory of p-adic fields, and Chapter 6 brings their applications to the study of algebraic number fields. We include here Shafare vich’s proof of the Kronecker-Weber theorem, and also the main properties of adeles and ideles.
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