Factorization: Unique and Otherwise

出版者:A K Peters

作者:Steven H. Weintraub

页数:250

出版时间:2008-5-10

价格:456.00

isbn号码:9781568812410

具体描述

Throughout the book, the exposition is crisp and self-contained, and Weintraub manages to strike a very nice balance between explicit computations and abstract theory. There are a good number of exercises, and plenty of directions one could go in after reading this book. . . . it approaches elementary number theory, a topic on which hundreds of books have been written, from a new direction. For that alone it should be rewarded, and this book has far more to offer."" -Darren Glass, MAA Reviews, November 2008

""This book offers an introduction to number theory bbuilt around the concept of unique factorization. After explaining the main players (integral domains and quadratic fields), the author proves that Euclidean rings are principal ideal domains, and that these have unique factorization. This is followed up with a lengthy discussion of examples of nonunique factorization in quadratic rings. ... The exposition is very detailed, and the examples and exercises take up more space than the actual text. Thus the text is well suited for self-study by motivated students, and even as a textbook for a first course in number theory..."" -Franz Lemmermeyer, Zentralblatt MATH, September 2009

""This very nice textbook starts with the fundamental theorem of arithmetic and heads directly to algebraic number theory presenting mainly results on quadratic number fields. ... Also Dirichlet's unit theorem is presented in a very understandable way. The book can be used as a first course in (algebraic) number theory. Many exercises lead to a deeper understanding."" -A. Winterhof, International Mathematical News, August 2009

""The starting point of this book is the concept of unique factorisation. Using an algebraic approach, the author ... opens the door to number theory up to the level of quadratic fields, together with a moderate introduction to algebraic number theory. ... The book can ... be used for self-study ... [and] ... can also be useful for instructors seeking an algebraically oriented complement for a standard text in elementary number theory."" -EMS Newsletter, December 2009

(mathematics, Lehigh U.) works through the concepts of factorization, an important feature of the system of natural numbers and their generalizations that can be written as a unique product of prime numbers and relates the ways in which factorization plays a key role in modern mathematics and its applications. After a fine introduction to basic notions, he covers unique factorization, the Gaussian integers, and Pell’s equation, and moves on to algebraic number theory. He also offers very good appendices on mathematical induction and congruences, sets of exercises for each chapter, and examples throughout. This is well-suited for a first course in number theory or for self-study by motivated readers down to the high school level."" -SciTech Book News, September 2008

""I do like [the author's] novel approach, and with the fact that this book is very nicely presented, with detailed explanations and many examples and exercises, it is safe to say that a first course in number theory following this book closely will be accessible and enjoyed by most second-year undergraduates and above."" -MathSciNet, November 2008

""I do like this novel approach, and with the fact that this book is very nicely presented, with detailed explanations and many examples and exercises, it is safe to say that a first course in number theory following this book closely will be accessible and enjoyed by most second-year undergraduates and above."" -P. G. Walsh, Mathematiacl Reviews, February 2009

""The concept of factorization, familiar in the ordinary system of whole numbers that can be written as a unique product of prime numbers, plays a central role in modern mathematics and its applications. This exposition of the classic theory leads the reader to an understanding of the current knowledge of the subject and its connections to other mathematical concepts . . . You will learn that instead of unique factorization being the norm and non-unique factorization the exception, the situation is reversed!"" -L'Enseignement Mathematique, December 2008

""In this concise, well-written book, Weintraub (Lehigh Univ.) wastes no time introducing the reader to the required concepts . . . Recommended."" -CHOICE Magazine , February 2009"

作者简介

Steven H. Weintraub was born in 1951, received his PhD in mathematics from Princeton University in 1974, and is currently Professor of Mathematics at Lehigh University. A specialist in areas of geometry, topology, and algebra, he is the author of approximately 50 research papers, and has visited and lectured at numerous universities around the world. This is his sixth book.

用户评价

阅读体验方面，这本书简直是一场智力上的“慢跑”，而非冲刺。作者的语言风格极其正式，充满了学术的庄重感，每一个句子都经过了精心的锤炼，生怕出现任何语义上的含糊不清。这在数学著作中是优点，但对于我来说，却有点像是在阅读一份厚厚的法律文书。我期待在“Unique and Otherwise”的对比中，能看到更多关于分解在不同数学分支中如何被灵活应用的案例研究。比如，分解理论在密码学中的实际应用，或者在代数几何中扮演的角色。这本书更侧重于“为什么”和“如何”在基础层面上成立，而不是“在哪里”和“在何种情境下”会发生变化。举个例子，对于某些特定的代数结构，分解可能不再是质数的累乘，而是更复杂的结构单元的组合。我本期待这本书能深入探讨这些现代化的、非传统的分解概念，并给出清晰的界限划分。但遗憾的是，大部分内容还是围绕着经典数论展开，对于那些拓宽了“分解”概念边界的现代理论，着墨甚少，仿佛视而不见。这本书的价值在于它的严谨，但它的局限也在于这种过度聚焦于传统定义的保守态度。

我必须承认，这本书在基础知识的梳理上做得相当到位，它为任何想要系统学习分解理论的人提供了一个极其坚实的地基。作者对基本概念的定义是无可挑剔的，逻辑链条的构建也相当流畅，基本上不会让你在阅读中因为概念的跳跃而感到迷失方向。这种循序渐进的教学方法，非常适合那些需要从零开始构建知识体系的读者。然而，当我读到关于“Otherwise”的部分时，我感受到了明显的力不从心。似乎作者对那些“不唯一”的情形，更多的是一种列举和描述，而缺乏对背后深层原因的剖析和理论体系的搭建。这种处理方式使得这部分内容显得有些单薄，仿佛只是为了呼应书名而勉强加入的注脚，而不是一个完整、深入的探讨。我期望看到的，是作者能以同等的热情和深度，去剖析那些打破传统唯一性规则的数学结构，挖掘出隐藏在“非唯一性”背后的新的数学规律。总的来说，这本书更像是一部制作精良的古典钢琴曲集，旋律优美，技巧娴熟，但缺少了一首能够打破沉闷、充满先锋精神的现代交响乐。它更侧重于“是这样”，而不是“也可以是那样”。

说实话，这本书的书名《Factorization: Unique and Otherwise》充满了诱惑力，它承诺了一场关于数学真理的辩论：何时分解是绝对的，何时它又变得模糊不清？我原本设想的是一场酣畅淋漓的逻辑搏杀，尤其是在处理那些超出传统整数范畴的抽象代数结构时。我特别关注那些“Otherwise”的部分——那些允许存在多个不同路径抵达“分解”终点的领域。想象一下，在某个特定的环（Ring）中，一个元素可以被写成A*B，也可以是C*D，而A, B, C, D之间又存在着复杂的关联，这种模糊性才是数学中最迷人的地方。然而，这本书给我的感觉是，它花了大约三分之二的篇幅来巩固“Unique”的部分，用大量篇幅去证明欧几里得的伟大，虽然这无可指摘，但却让我对书名所暗示的“不唯一”的精彩探索感到意犹未尽。那些更具挑战性的例子，那些需要跳出常规思维框架才能理解的分解悖论，它们像是被巧妙地藏在了厚厚的序言之后，需要读者费力去挖掘。叙述风格上，作者的笔触非常严谨，达到了教科书级别的精确性，但这牺牲了一定的可读性和趣味性。如果你只是想巩固数论的经典基础，这本书是绝佳的选择，但如果你期待的是一场关于分解本质的颠覆性认知之旅，你可能会觉得它有些过于保守和循规蹈矩了。

我最近读了很多关于结构化数学的著作，这本书给我的体验是独树一帜的。它的结构设计得极其精巧，仿佛一座迷宫，但迷宫的墙壁是由清晰的数学定义和无可辩驳的证明构筑而成的。最让我印象深刻的是它对于“过程”的强调，而不仅仅是结果。作者似乎对数字是如何被分解的那个“中介状态”抱有近乎痴迷的热情。比如，在讨论高斯整数或其他特定数域时，他不仅仅给出了分解式，还详细剖析了每一步操作背后的代数动机，这使得即便是相对简单的分解过程，也充满了洞察力。这种细致入微的讲解，对于初学者来说无疑是宝贵的财富，它能帮助他们建立起对分解操作的直觉。然而，对于我这种更关注理论前沿的读者来说，这种深入到每一个细节的讲解，导致了整体阅读进度的缓慢。我更倾向于那些能快速建立起抽象框架，然后让我自己去填补细节的著作。这本书更像是那位耐心无比的导师，手把手地教你如何走路，而不是给你一张地图，让你自己去探索广袤的未知领域。它的力量在于其无懈可击的稳固性，但代价是，它没能提供那种令人振奋的、突破边界的“飞跃感”。

这本书，说实话，我抱着非常高的期望走进来的，毕竟“分解”这个主题在数学领域里简直是永恒的魅力所在，它不仅关乎数论那严丝合缝的逻辑推演，更渗透在代数结构、甚至信息安全等更广阔的天地。我期待的是一次对“唯一性”的深度哲学探讨，而不是那种仅仅停留在高中代数课本上的简单乘法拆解。当我翻开第一章时，首先吸引我的是作者那种近乎老派的、对数学美感的执着。他似乎不急于抛出复杂的定理，而是用一种近乎诗意的语言去描绘数字世界的底层结构，就像一位技艺精湛的钟表匠，耐心地向你展示每一个齿轮是如何咬合在一起的。然而，随着阅读的深入，我开始感到一丝困惑。书中似乎花了过多的篇幅去铺陈一些基础概念的定义和历史溯源，这些内容虽然扎实，但对于一个已经有一定数学背景的读者来说，显得有些冗余和拖沓。我更希望看到的是那些在现代研究前沿那些“非唯一”分解的可能性空间，比如在某些非经典代数结构中，分解的边界是如何被模糊和重塑的。这本书的叙事节奏，就像一列老式的蒸汽火车，起步缓慢，虽然最终能到达目的地，但沿途的风景略显单调，缺乏那种让人心跳加速的、思想火花的瞬间碰撞。整体而言，它更像是一部详尽的、注重基础的教材，而非一场激动人心的思想探险。