# From Polynomial to Modern Algebra

Examples Polynomial with degree > 5

# Abstract “Nonsenses” in Abstract Math make “Sense”

After 40 years of learning Abstract Algebra (aka Modern Math yet it is a 200-year-old Math since 19CE Galois invented Group Theory), through the axioms and theorems in math textbooks and lectures, then there is an Eureka “AHA!” revelation when one studies later the “Category Theory” (aka “Abstract Nonsense”) invented only in 1950s by 2 Harvard professors.

A good Abstract Math teacher is best to be a “non-mathematician” , who would be able to use ordinary common-sense concrete examples to explain the abstract concepts: …

Let me explain my points with the 4 Pillars of Abstract Algebra :

$\boxed {\text {(1) Field (2) Ring (3) Group (4) Vector Space}}$

Note: the above “1-2-3 & 4″ sequence is a natural intuitive learning sequence, but the didactical / pedagogical sequence is “3-2-1 & 4″, that explains why most students could not grasp the philosophical essence of Abstract Algebra, other than the “technical” axioms & theorems.

If a number system (Calculator arithmetic) has 4 operations (+ – * ÷ ), then it is a “Field” (域) – Eg. Real, Complex, Z/pZ (Integer mod Prime)…

If a number system with +, – and * (but no ÷), then it is a “Ring” (环).
eg1. Clock arithmetic {1,2, 3,…,12} = Z/12 (note: 12 is non-prime). [Note: the clock shape is like a ring, hence the German called this Clock number a “Ring”.]
eg2: Matrix (can’t ÷ matrices)
eg3. Polynomial is a ring (can’t ÷ 0 which is also a polynomial).

If a system (G) with 1 operation (○) and a set of elements {x y z w …} that is “closed” (kaki-lang 自己人, any 2 elements x ○ y = z still stay inside G ) , associative (ie bracket orderless) :(xy)z = x(yz), a neutral element (e) s.t. x+e = x = e+x, and inversible ($x^{-1}$, $y^{-1}$ … still inside G), then G is a Group.
eg. {Integer, +}: 2’s inverse (-2), neutral 0, (2+3)+4=2+(3+4)
eg2. Triangle rotation 120 degree & flip about 3 inner axes.

If a non-empty system V ={v u w z …} that is “closed”if any of its 2 elements (called vectors v, u) v + u = w still in V,
AND if any vector multiply it by a scalar “λ” s.t. “λv” still in V, then V is a Vector Space (向量空间)。
eg1. Matrix (Vector) Space
eg2. Function (Vector) Space
eg3. Polynomial (Vector) Space

Summarise the above 4 or more systems into 1 Big System called “Category” (C) 范畴, then study relation (arrow or morphism) between f: C1 -> C2, this is “Category Theory“.

In any number system (aka algebraic structure), you can find the “Yin / Yang” (阴阳) duality : eg. “Algebra” [#] / “Co-Algebra”, Homology (同调) / Co-Homology (上同调)… if we find it difficult to solve a problem in the “Yang”-aspect. eg. In “Algebraic Topology”: “Homology” (ie “Holes”) with only “+” operation, then we could study its “Yin”-aspect Co-Homology in Ring structure, ie with the more powerful “*” multiplicative operation.

Note [#]: “Algebra” (an American invented structure) is a “Vector Space” plus multiplication between vectors. (Analogy in Physics : Cross Product of vectors).

# 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:

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]