Advanced Algebra (Third Edition), Volume II
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Summary
This book is the second volume of Qiu Weisheng's Advanced Algebra, Third Edition, and together with the first volume it forms a complete textbook for an advanced algebra course. Whereas the first volume follows the concrete objects of linear algebra, the second turns to abstract structural content: throughout, the book is informed by the modern mathematical viewpoint of structures and morphisms (maps that preserve operations), seeking to train students in the basics and fundamental methods while cultivating the habit of "thinking mathematically." Volume II comprises four chapters. Chapter 7, "Polynomial Rings," begins with the concept and universal properties of the polynomial ring in one variable, then discusses divisibility and division with remainder, greatest common divisors, irreducible polynomials and the unique factorization theorem, repeated factors, and the irreducibility and factorization of polynomials over the complex, real, and rational fields, closing with polynomial rings in several variables, symmetric polynomials, and the residue class ring modulo m and the characteristic of a field. Chapter 8, "Linear Spaces," treats the structure of linear spaces, the intersection, sum, and direct sum of subspaces, and the isomorphism and quotient spaces of linear spaces. Chapter 9, "Linear Maps," covers linear maps and their operations, kernel and image, and matrix representation; eigenvalues and eigenvectors of linear transformations, conditions for diagonalizability, invariant subspaces, the Cayley-Hamilton theorem, minimal polynomials, the structure of nilpotent transformations, and the Jordan canonical form; and linear functions and dual spaces. Chapter 10, "Linear Spaces with Metrics," starts from bilinear functions and proceeds through Euclidean spaces, orthogonal complements and orthogonal projections, orthogonal and symmetric transformations, extending to unitary spaces, unitary transformations, Hermitian transformations, and orthogonal and symplectic spaces. Exercises are provided section by section, answers and hints appear at the end of the book, and several reading passages are interspersed. The book can serve as a textbook for advanced algebra courses at comprehensive universities, universities of science and engineering, and teachers colleges.
Editor's review
在众多高等代数教材中，丘维声这本的鲜明之处在于“结构—映射”的现代观点。作者把线性空间、线性映射放在舞台中心，再让矩阵、标准形、度量等对象依次登场，读者学到的不是一堆互不相连的计算技巧，而是一套可以延伸到抽象代数、泛函分析与几何的整体框架。 下册尤其见功力：多项式环一章把整除、因式分解与域的特征放在同一逻辑链条上，为后续学习抽象代数预留接口；线性映射一章从核与象、不变子空间一路推到最小多项式、幂零分解与若尔当标准形，证明线索清晰，几乎每一步都有习题配合；第10章把欧氏空间与酉空间、正交空间、辛空间并置比较，让“度量”成为统一视角。穿插书中的阅读材料把有限域、线性码、结式等周边话题自然地引入，既不打断主线，又能激发兴趣。作者四十余年教学经验的沉淀体现在难度梯度上：正文可读，习题成体系，答案与提示附于书末，自学、复习均可依托。若你希望高等代数不只是考研工具，而是真正的数学训练，这本书值得长期放在案头。
Who it's for
A good fit for
- 数学类、统计学类专业本科生，作为高等代数或线性代数课程的主教材
- 准备攻读数学、计算机、自动化等方向研究生、需系统复习代数基础的考生
- 希望深入理解若尔当标准形、对偶空间、酉空间等较难内容的理工科高年级学生
- 讲授高等代数课程、需要补充阅读材料与习题资源的高校教师
Not a good fit for
- 完全没有线性代数基础、只想先了解矩阵运算入门的自学者
- 只需要一两周速成应付考试、不打算读证明与定理推导的读者
- 高中数学竞赛之外、对抽象证明兴趣有限的中学阶段读者
Reading tips
- 务必先通读同版本上册（线性方程组、行列式、矩阵、二次型等），下册的不少例子与符号沿用上册约定。
- 每节先独立做习题再核对书末提示；若某题卡住超过半小时，记下思路盲点，读下一节后再回看往往豁然开朗。
- 第9章与第10章建议对照上册的矩阵相抵、相似与合同来读，把“具体计算”与“结构定理”一一挂钩。
- 穿插的阅读材料（有限域、线性码、正交变换等）可按兴趣跳读，不影响主线推进。
Contents
- Chapter 7 Polynomial Rings
- §1 The Concept of the Polynomial Ring in One Variable and Its Universal Properties
- §2 Divisibility, Division with Remainder
- Reading Passage 1: Division with Remainder in the Ring of Integers
- §3 Greatest Common Divisors
- Reading Passage 2: Greatest Common Factors
- §4 Irreducible Polynomials, the Unique Factorization Theorem
- Reading Passage 3: The Fundamental Theorem of Arithmetic
- §5 Repeated Factors
- §6 Roots of Polynomials, Irreducible Polynomials over the Complex Field
- §7 Irreducible Polynomials over the Real Field
- §8 Irreducible Polynomials over the Rational Field
- §9 Polynomial Rings in Several Variables
- §10 Symmetric Polynomials
- §11 The Residue Class Ring Modulo m, Fields, the Characteristic of a Field
- Chapter 8 Linear Spaces
- §1 The Structure of Linear Spaces
- §2 Subspaces and Their Intersection and Sum, Direct Sums of Subspaces
- §3 Isomorphism of Linear Spaces
- Reading Passage 4: The Number of Elements of a Finite Field
- §4 Quotient Spaces
- Reading Passage 5: Linear Codes
- Chapter 9 Linear Maps
- §1 Linear Maps and Their Operations
- §2 Kernel and Image of a Linear Map
- §3 Matrix Representation of Linear Maps
- Reading Passage 6: Conditions for Pairwise Orthogonal Idempotent Transformations
- §4 Eigenvalues and Eigenvectors of Linear Transformations, Conditions for Diagonalizability
- §5 Invariant Subspaces of Linear Transformations
- §6 The Cayley-Hamilton Theorem
- §7 Minimal Polynomials of Linear Transformations
- §8 The Structure of Nilpotent Transformations
- §9 The Jordan Canonical Form of Linear Transformations
- §10 Linear Functions and Dual Spaces
- Chapter 10 Linear Spaces with Metrics
- §1 Bilinear Functions
- §2 Euclidean Spaces
- §3 Orthogonal Complements, Orthogonal Projections
- §4 Orthogonal and Symmetric Transformations
- Reading Passage 7: Matrix Representation of the Simplest Form of Orthogonal Transformations
FAQ
这本书是上册、下册还是合订本？
这是丘维声《高等代数》第三版的下册，单独成册，内容包括多项式环、线性空间、线性映射、具有度量的线性空间四章，不含上册的线性方程组、行列式与矩阵等章节。
第三版和第四版下册有什么区别？
第四版为2026年出版，开本、字数与部分小节有调整，例如增加了结式、线性变换的有理标准形等阅读材料，篇幅更大；第三版下册则较为精简，两版主体框架一致。选购时请核对封底版次与ISBN。
书里的习题有答案吗？
书末附有“习题答案与提示”，按节给出参考答案或解题提示；阅读材料类内容一般以拓展为主，不完全配套完整解答。
适合用来自学吗？
整体可读性较好，证明步骤较细，但下册内容偏抽象，自学时最好先掌握上册与基本的矩阵运算，并配合其他参考书与课程视频使用。
“若尔当标准形”在哪一章？
在第9章“线性映射”§9，此前先讲了可对角化条件、不变子空间、哈密顿—凯莱定理、最小多项式与幂零变换的结构，属于层层递进的安排。
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About the author
Qiu Weisheng
Author · Chinese · 1945
Qiu Weisheng, born in 1945 in Shanghang, Fujian, is a professor and doctoral supervisor at the School of Mathematical Sciences, Peking University. His research focuses on algebraic combinatorics, group representation theory, and cryptography, and he has devoted considerable effort to mathematics teaching and textbook development, authoring textbooks such as Advanced Algebra and Analytic Geometry. His writing is known for clear concepts and rigorous reasoning, guiding readers from definitions and structures to an understanding of the connections between geometric objects and algebraic methods; his textbooks are widely used in university mathematics courses and exam preparation. In 2003 he received the first National Teaching Master Award, reflecting his sustained contributions to mathematics education. Publicly available information on his specific career details is limited, and existing accounts center mainly on his academic directions and textbook achievements.
