基础数论

作者：（美）杜德利著，周仲良译

出版社：哈尔滨工业大学出版社

出版时间：2011-3-1

版次：1页数：230字数：273000

印刷时间：2011-3-1开本：16开纸张：胶版纸

印次：1ISBN：9787560332048包装：平装

杜德利所著的《基础数论》对初等数论的大多数论题进行了介绍。第一章至第五章中，推导了整数和同余武的基本性质，第六章给出了费马定理和威尔逊定理的证明，第七章至第九章介绍了数论函数d，σ和φ，在第十章至第十二章中，推出了重要的二次互反性定理。接下去是多少有点互不相关的三部分材料：关于数的表示式(第十三章至第十五章)，丢番图方程(第十六章至第二十章)和素数(第二十一章至第二十二章)。我认为，在数论中，习题和练习特别重要，也很有趣，因此，在第二十三章中收进了105道杂题。它们大致上是按照难度而未考虑论题排列起来的。

杜德利所著的《基础数论》对初等数论的大多数论题进行了介绍。推导了整数和同余式的基本性质，给出了费马定理和威尔逊定理的证明，介绍了几个数论函数以及丢番图方程和素数等知识，推出了重要的二次互反性定理。全书共收进了一千多道练习和习题，且练习插在文(和一些证明)中，习题则附在各章末尾。

《基础数论》适用于高等学校数学类专业作为教材使用，也适用于对数学特别是数论知识感兴趣的读者使用。

第一章整数

第二章因子分解的唯一性

第三章线性不定方程

第四章同余式

第五章线性同余式

第六章费马定理和威尔逊定理

第七章整数的因子

第八章完全数

第九章欧拉定理和欧拉函数

第十章原根和指数

第十一章二次同余式

第十二章二次互反性

第十三章用不同的基表示的数

第十四章十二进位数

第十五章十进位小数

作者：(美) 杜德利 译者：周仲良

The first part of this volume is based on a course taught at PrincetonUniversity in 1961-62; at that time, an excellent set of notes was preparedby David Cantor, and it was originally my intention to make these notesavailable to the mathematical public with only quite minor changes.Then, among some old papers of mine, I accidentally came across along=forgotten manuscript by Chevalley, of pre-war vintage （forgotten,that is to say, both by me and by its author） which, to my taste at least,seemed to have aged very well. It contained a brief but essentially com-plete account of the main features of classfield theory, both local andglobal; and it soon became obvious that the usefulness of the intendedvolume would be greatly enhanced if I included such a treatment of thistopic. It had to be expanded, in accordance with my own plans, but itsoutline could be preserved without much change. In fact, I have adheredto it rather closely at some critical points.
To improve upon Hecke, in a treatment along classical lines of thetheory of algebrai~ numbers, would be a futile and impossible task. Aswill become apparent from the first pages of this book, I have rathertried to draw the conclusions from the developments of the last thirtyyears, whereby locally compact groups, measure and integration havebeen seen to play an increasingly important role in classical number-theory. In the days of Dirichlet and Hermite, and even of Minkowski,the appeal to "continuous variables" in arithmetical questions may wellhave seemed to come out of some magician's bag of tricks. In retrospect,we see now that the real numbers appear there as one of the infinitelymany completions of the prime field, one which is neither more nor lessinteresting to the arithmetician than its p=adic companions, and thatthere is at least one language and one technique, that of the adeles, for bringing them all together under one roof and making them cooperate for a common purpose. It is needless here to go into the history of thesedevelopments; suffice it to mention such names as Hensel, Hasse, Chevalley, Artin; every one of these, and more recently Iwasawa, Tate, Tamagawa, helped to make some significant step forward along this road. Once the presence of the real field, albeit at infinite distance, ceases to be regarded as a necessary ingredient in the arithmetician's brew.