# Pre-requisites For Abstract Algebra

The 2 important pre-requisites for Abstract Algebra are “abstract thinking”, namely :

1. You must not think of “concrete” math objects (geometrical shapes, Integer, Real, complex numbers, polynomials, matrices…), but rather their “generalised” math objects (Group, Ring, Field, Vector Space…).
2. Rigourous Proof-oriented rather than computation-oriented.

The foundation of Abstract Algebra is “Set Theory”, make effort to master the basic concepts : eg.

• Sub-Set,
• Equivalence Relation (reflexive, symmetric, transitive),
• Partitioning, Quotient Set, Co-Set
• Necessary condition (“=>”), Sufficient condition (“<=”). Both conditions (“<=>”, aka “if and only if”)
• Proof : “A = B” if and only if a A, a B => A B, and, b B, b A =>B A [我中有你, 你中有我 <=>你我合一]
• Check if a function is “well-defined”. (定义良好)
• These concepts / techniques repeat in every branch of Abstract Algebra which deals with all kinds of “Algebraic Structures”, from Group Theory to Ring Theory to Field Theory … to (Advanced PhD Math) Category Theory – aka “The Abstract Nonsense”.

# 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

THEOREMS(PROOF Here)

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

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)。

———————

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:

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]

# “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)