Why call the Algebraic Structure Z a “Ring” ?

    Integer Z (from german “Zahlen”) is a Ring (环). It has + – × but not ÷ (otherwise it would become a fraction outside  Z).

From the above 8 Ring axioms, we can derive the other rule:
Prove \boxed {a.0 = 0}
a.1 = a. (1+0) [identity Law]
a.1 = a.1 + a.0 [distributive law]
add (-a.1) to both sides,
(-a.1) + a.1 = (-a.1) + a.1 + a.0 [inverse, associative  law ]
0 = a.0 [QED]

The Finite Ring (Zn) is best illustrated by Z12 or “Clock Number” {1, 2, 3, …11, 12=0} which forms a “Ring” – hence the name “Ring” (环).

An important property of “special ring” Zp for any prime number p, eg. Z2 = {0,  1} has additional “÷” operation (or multiplicative inverse) besides {+ – ×}, so it is a “Field”.

In Z2, 1.1 = 1 (mod 2) <=> 1 is multiplicative inverse of itself.

Zp is useful in Encryption “RSA” algorithm using Prime number theorems such as The Chinese Reminder Theorem and Fermat’s Little Theorem.

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 solution  to solve algebraic equations代数方程式.

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}}}}

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

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

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

代 数拓扑 Algebraic Topology (Part 1/3)

Excellent Advanced Math Lecture Series (Part 1 to 3) by 齊震宇老師

(2012.09.10) Part I:

History: 1900 H. Poincaré invented Topology from Euler Characteristic (V -E + R = 2)

Motivation of Algebraic Topology : Find Invariants [1]of various topological spaces (in higher dimension). 求拓扑空间的“不变量” eg.

  • Vector Space (to + – , × ÷ by multiplier Field scalars);
  • Ring (to + x) in co-homology
  • etc.

then apply algebra (Linear Algebra, Matrices) with computer to compute these invariants  (homology, co-homology, etc).

A topological space can be formed by a “Big Data” Point Set, e.g. genes, tumors, drugs, images, graphics, etc. By finding (co)- / homology – hence the intuitive Chinese term (上) /同调 [2] – is to find “holes” in the Big Data in the 10,000 (e.g.) dimensional space the hidden information (co-relationship, patterns, etc).
Note: [1] Analogy of an”Invariant” in Population: eg. “Age” is an invariant can be added in the “Population Space” as the average age of the citizens.

Side Reading (Very Clear) : Invariant and the Fundamental Group Primer

Note [2]: Homology 同调 = same “tune”.

南朝 刘宋 谢灵运山水诗:
同调 = Homology
(希腊 homo = 同, -logy = 知识 / 调)

– “Reading an ancient text  allows us to think “in tune” (or resonant) with the ancient author.”

[温习] Category Theory Foundation – 3 important concepts:

  • Categories
  • Functors
  • Natural Transformation

[Skip if you are familiar with Category Theory Basics: Video 16:30 mins to 66:00 mins.]

[主题] Singular Homology Groups 奇异同调群  (See excellent writeup in Wikipedia) (Video 66:20 mins to end)

  1. Singular Simplices
  2. Singular Chain Groups
  3. Boundary Operation
  • Singular Chain Complex:
  • Part 1/3 Video (Whole) :

    Russian & World Math Education

    ​In the world of Math education there are 3 big schools (门派) — in which the author had the good fortune to study under 3 different Math pedagogies:

    “武当派” French (German) ,  “少林派” Russian  (China) ,  “华山派” UK (USA).

    ( ) : derivative of its parent school. eg. China derived from Russian school in 1960s by Hua Luogeng.

    武当派 : 内功, 以柔尅刚, 四两拨千斤 <=> “Soft” Math, Abstract, Theoretical, Generalized.

    少林派: 拳脚硬功夫 <=> “Hard” Math, Algorithmic.

    华山派: 剑法轻灵 <=> “Applied” Math, Astute, Computer-aided.

    The 3 schools’ pioneering grand masters (掌门人) since 16th century till 21st century, in between the 19th century (during the French Revolution) Modern Math (近代数学) is the critical milestone, the other (现代数学) is WW2 : –

    France: Descartes / Fermat / Pascal  (17 CE : Analytical Geometry, Number Theory, Probability), Cauchy / Lagrange / Fourier /Galois  (19 CE, Modern Math : Analysis, Abstract Algebra), Poincaré (20CE Polymath, Topology), André Weil (WW2,  Bourbaki school), Alexander Grothendieck (21 CE, Differential Geometry)…

    Note: 1/3 of Fields Medals won by French.

    (Germany): Leibniz (18CE, Calculus, Binary Algebra). Felix Klein (20 CE, learnt from French master Camile Jordan) established the Gottingen school (World Center of Math before WW2, destroyed by Hitler). Successors: Gauss ( Polymath), Hilbert , Riemann (Prime Number) , Cantor  (Infinity Math), Emile Noether  (Axiomatic Algebra, Ring Theory) …

    Note: Swiss-German branch – Bernouille father & 3 sons and student Euler (Polymath).

    RussiaА. Н. Колмогоров (20 CE, Polymath), Grigori Perelman (21 CE, Topology, who rejected Fields medal).

    (China): 华罗庚 (Hua Luogeng, Number Theory), 陈省身 (SS Chern, China/USA, Differential Geometry),  陈景润 (“Chen Theorem”, Number Theory), 丘成桐 (ST Yau, HK/USA, Differential Geometry), 吴文俊 (Wu Wenjun, Machine Proof of Geometry )。

    UK: Isaac Newton  (18 CE,  Calculus), Hardy / Littlewood / Ramanujian (Number Theory), Bertrand Russell  (Logic), Andrew Wiles (21 CE, proved 380-year-old Fermat’s Last Theorem).

    (USA): 20CE Godël (Austrian / USA), Paul Erdös (Hungary / USA), Eilenberg /MacLane (WW2, Category Theory).

    Note: Terence Tao (陶轩哲 Australia / USA, 21 CE, Number Theory)


    拓扑学 Topology (Part 1)

    3-Sphere = 2 dimensional 
    (站在locale 看周围 neighbourhood) 
    eg. 站在地球一点上, 看脚下似平面 (2-D)

    Differentiable 可微性 ( => 连续性 continuity ) = 平滑 (smooth) 线性变化 [Intuitively]

    全微分 {f: R^{2} \to R^{1}}
    represented by matrix (Linear Transformation):
    { \big[\frac {\partial f } {\partial x} , \frac {\partial f } {\partial y}\big] } .\begin{pmatrix} x\\ y \end{pmatrix} = { \big[\frac {\partial f } {\partial x}.x + \frac {\partial  f } {\partial y}.y\big]}

    f: R^{n} \to R^{m} => Matrix (n,m)

    微分 = 线性变化
    \boxed { \text {Differentiation} = \text{ Linear Transformation} }

    Find “mini – max ” <=> Kernel { \big[\frac {\partial f } {\partial x} , \frac {\partial f } {\partial y}\big] } .

    梯度 (gradient): 地形图 等高线 f(x,y)

    垂直方向 = { \big(\frac {\partial f } {\partial x}.i + \frac {\partial f } {\partial y}.j\big)}

    路人行走的路线: { \big[\frac {\partial g} {\partial x}, \frac {\partial g } {\partial y}\big]}

    路线 g(x,y) 的 (max.)极大值 if and only if

    f 和 g 的梯度平行: \iff
    { \boxed { \bigg(\frac {\partial f } {\partial x}.i + \frac {\partial f } {\partial y}.j\bigg)= \lambda. \bigg(\frac {\partial g } {\partial x}.i + \frac {\partial g} {\partial y}.j\bigg) }}

    Lagrange Multiplier : \lambda