Integral Domains 整环 (Abstract Algebra)

Remember when you cancel a common factor at both sides of an equation, you must check if the factor is non-zero, otherwise you would miss some answers.

This is about Cancellation Law, related to few Number Theory Properties : 

  • Zero Divisors,
  • Integral Domain.

Origin of “Integral” => Integers

Definition of Integral Domian:

Property: Cancellation Law  



  1. Every Field is an Integral Domain.
  2. Every finite Integral Domain is a Field

北大 高等代数 (1) Beijing University Advanced Algebra

辛弃疾的《青玉案·元夕》:“…众里寻他千百度;蓦然回首,那人却在灯火阑珊处。” –表达出了我的一种 (网上)意外相逢的喜悦,又表现出对心中(名师)的追求。

2011 年 北京大学教授 丘维声教授 被邀给清华大学 物理系(大学一年级) 讲一学期课 : (Advanced Algebra) 高等代数, aka 抽象代数 (Abstract Algebra)。

丘维声(1945年2月-)生于福建省龙岩市,中国数学家、教育家。16岁时以全国高考状元的成绩考入北京大学,1978年3月至今担任北京大学数学科学学院教授,多年坚持讲授数学专业基础课程。截至2013年,共著有包括《高等代数(上册、下册)》、《简明线性代数》两本国家级规划教材在内的40部著述。于1993-97年的一系列文章中逐步解决了n=3pr情形的乘子猜想,并取得了一系列进展。[Ref: Wekipedia ]


72岁的丘教授学问渊博, 善于启发, 尤其有别于欧美的”因抽象而抽象”教法, 他独特地提倡用”直觉” (Intuition) – 几何概念, 日常生活例子 (数学本来就是源于生活)- 来吸收高深数学的概念 (见: 数学思维法), 谆谆教导, 像古代无私倾囊相授的名师。

全部 151 (小时) 讲课。如果没时间, 建议看第1&第2课 Overview 。

第一课: 导言 : n 维 方程组 – 矩阵 (Matrix)- n 维 向量空间 (Vector Space) – 线性空间 (Linear Space)


上表 (左右对称): 

左。双线性函数 (Bi-linear functions) – eg.內积 (Inner Product x.y = |x|.|y|.cos A)

右:  线性映射 (Linear Map) – 保存 线性 (linearity  ie + 法 , 数乘法 scalar multiplication)

线性 : 一切 平面上的, 或球体表面的点平面 (如: 地球的某点是平地)

线性空间 + 度量 norm =>

  • Euclidean Space (R) => (正交 orthogonal , 对称 symmetric) 变换
  • 酉空间 Unitary Space (C)…  => 变换, Hermite变换

近代代数 (Modern Math since 19CE  Galois) : 从 研究 结构 (环域群) 开始: Polynomial Ring, Algebraic structures (Ring, Field, Group).

第三课: 简化行阶梯形矩阵 Reduced Row Echelon Matrix

第四课: 例子 (无解)

第五课: 证明 无解/唯一解/无穷解, 行列式 (Determinant, Det)
[几何直觉]: 任何2线 1) 向交(唯一解) ; 2) 平行 (无解) ; 3) 重叠 (无穷解)。


: O = d Det = 0

  • Rank r < n : 无穷解Det = 0
  • Rank r  = n : 唯一解 Det \neq 0

继续: n阶 行列式 

In Search for Radical Roots of Polynomial Equations of degree n > 1

Take note: Find roots (根) to solve polynomial (多项式方程式) equations, but find solutions (解) to solve simultaneous equations (联式方程式).

Radical : (Latin Radix = root): \sqrt [n]{x}

Quadratic equation (二次方程式) [最早发现者 : Babylon  和 三国时期的吴国 数学家 赵爽]

{a.x^{2} + b.x + c = 0}

\boxed{x= \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}}

Cubic Equation: 16 CE Italians del Ferro,  Tartaglia & Cardano
{a.x^{3} = p.x + q }

Cardano Formula (1545 《Ars Magna》):
\boxed {x = \sqrt [3]{\frac {q}{2} + \sqrt{{ (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}} + \sqrt [3]{\frac {q}{2} -\sqrt{ { (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}}}

{x^{3} = 15x + 4}
By obvious guess,  x = 4
Using Cardano formula,
x = \sqrt[3]{2+ 11 \sqrt{-1}} + \sqrt [3]{2 - 11 \sqrt{-1}}

They discovered the first time in history the “Imaginary” number (aka Complex number):
\boxed {i = \sqrt{-1}}
(2 + i)^{3} =2+11i
(2 - i)^{3} =2-11i
x = (2 + i) + (2 - i) = 4

Quartic Equation: by Cardano’s student Ferrari
{a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}

Quintic Equation:
{a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}

No radical solution (Unsolvability) was suspected by Ruffini (1799), proved by Norwegian Abel (1826), but explained by French 19-year-old boy Évariste Galois (discovered in 1831, published only after his death in 1846) with his new invention : Abstract Algebra “Group“(群) & “Field” (域)。


Group Theory is Advanced Math.
Field Theory is Elementary Math.

Field is the Algebraic structure which has 4 operations on calculator (+ – × ÷). Examples : Rational number (\mathbb{Q}), Real (\mathbb{R}), Complex (\mathbb{C}), \mathbb{Z}_{p}  (Integer modulo prime, eg.Z2 = {0, 1}) , etc.

If \mathbb{Q}   (“a”, “b”) is adjoined with irrational (eg. \sqrt {2})  to become a larger Field (extension) \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2}
it has a beautiful “Symmetry” aka Conjugate
(a - b\sqrt {2}) 

Field Extension of \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} :

Any equation P(x) = 0
with root in \mathbb{Q} (\sqrt {2}) = a +b\sqrt {2} will have
another conjugate root (a - b\sqrt {2})

Galois exploited such root symmetry in his Group structure to explain the unsolvability for polynomial equations of quintic degree and above.

Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]

Modern Algebra (Abstract Algebra) Made Easy

UReddit Courses:

Modern Algebra (Abstract Algebra) Made Easy

This video series is really well done ! short and sharp, yet cover the entire syllabus in the Group Theory.

Strongly recommended for those Math-inclined students from upper secondary schools (Secondary 3 to JC2). Although the Singapore school Math syllabus based on Cambridge ‘O’ and ‘A’ level do not cover modern math – which is a serious weakness for being biaised on computational applied math, an outdated pedagogy for the last 40 years with no major changes – we miss the latest Math development since 19 century, the so called ‘Modern Math’ but already 300 years old.

Group Theory is the stepping stone to open the door of interesting advanced Math, physics, chemistry, bio-science and engineering. It should not be limited only to the Math-major undergraduates in university. (Note: Why France and China make Modern Algebra compulsory for all science and engineering students )

Part 0: Binary Operations

Part 1: Group

Note: Why ‘e’ for Identity, ‘Z’ integers

Part 2: Subgroup

Part 3: Cyclic Group & its Generator

Part 4: Permutations

Part 5: Orbits & Cycles

Part 6 : Cosets & Lagrange’s Theorem

Part 7 : Direct Products / Finitely generated Abelian groups

Part 8: Group Homomorphism

Part 9: Quotient Groups

Part 10: Rings & Fields

Part 11: Integral Domains

Reference: further studies in deeper and advanced Abstract Algebra at:

Harvard Online Free Course by Prof Benedict Gross