# 北大 高等代数 (1) Beijing University Advanced Algebra

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)。

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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。

http://www.bilibili.com/mobile/video/av7336544.html?from=groupmessage

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

[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)。

n次方程組的解也只有3个情况:

: O = d $Det = 0$

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

# 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》):
$\boxed {x = \sqrt [3]{\frac {q}{2} + \sqrt{{ (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}} + \sqrt [3]{\frac {q}{2} -\sqrt{ { (\frac {q}{2})}^{2} - { (\frac {p}{3})}^{3}}}}$

Example:
${x^{3} = 15x + 4}$
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):
$\boxed {i = \sqrt{-1}}$
then
$(2 + i)^{3} =2+11i$
$(2 - i)^{3} =2-11i$
$x = (2 + i) + (2 - i) = 4$

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)。

Notes:

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]

# “Well-defined”( “定义良好”)

“定义良好” (Well-defined)

Prove : f is well-defined ?

# 抽象代数 Abstract Algebra

1. Euclid 5条公理 (Axioms) => 全部 几何 (Geometry)
2. Galois 运算律 (Laws of Operations) => 抽象代数 (Abstract Algebra)

# Modern Algebra (Abstract Algebra) Made Easy

UReddit Courses:
http://ureddit.com/category/23446/mathematics-and-statistics

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

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