The birth of Abstract Algebra :

Galois “Group Theory” – >

Dedekind “Field Theory” ->

Noether (Ring / Ideal)

-> Noether Axiomatic Abstract Algebra

# Tag Archives: Abstract Algebra

# Quora: Is Abstract Math a fluke (偶然) ？

# 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 :

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 (, … 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).

Reference:https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6

# Pre-requisites For Abstract Algebra

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

- 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…). **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”
∀a ∈ A, a ∈ B => A ⊂ B, and, ∀b ∈ B, b ∈ A =>B ⊂ A [*if and only if**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

**Reading**:

**THEOREMS**: (PROOF Here)

- Every
**Field**is an**Integral Domain**. - 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 。

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

**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)**

第二课**: **

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)…****=>****酉****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)。

n次方程組的解也只有3个情况:

**
无解**:

**O = d**

有解:

**Rank r < n :****无穷解****Rank r = n :****唯一解**

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):

**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]

**Cubic Equation**: 16 CE Italians del Ferro, Tartaglia & Cardano

Cardano Formula (1545 《Ars Magna》):

**Example**:

By obvious guess, x = 4

Using Cardano formula,

They discovered the first time in history the *“Imaginary” number* (aka **Complex** number):

then

Quartic Equation: by Cardano’s student Ferrari

**Quintic Equation**:

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 , Real , Complex , (Integer modulo prime, eg.Z2 = {0, 1}) , etc.

If (“a”, “b”) is adjoined with irrational (eg. ) to become a **larger Field (extension) **

it has a beautiful “**Symmetry**” aka **Conjugate**

Field Extension of :

Any equation P(x) = 0

with root in will have

another **conjugate** root

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]

# Monomorphism (Injective Homomorphism)

# “Well-defined”( “定义良好”)

# 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**

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**

# Structures algébriques 1 (Introduction : Monoïde…)

# Harvard Online Course: Abstract Algebra

Prof Benedict Gross is one of the best Algebra professors I have seen – he can explain so well the abstract concepts, without injecting fear and confusion to the students.

As Prof Gross had brilliantly said in the beginning of this Lecture 1:

**Algebra is the language of Math**.

Since Math is the language of science,

therefore any serious Science needs to speak in Algebra language.

Today, if you read a research paper on any math (or Computer Science, Mathematical Physics…) topic, hardly you can avoid these “basic” algebraic lingoes: Group, Ring, Field, Vector Spaces, Quotient Group, Ideal, …

I strongly recommend to anyone who likes to study Modern Algebra but afraid of the abstractness, this is the course (free) for you. I can guarantee you by the halfway (15th lecture) you will have a solid foundation, and by the last lecture you will be able to follow high-level math lectures — because you understand the language of Math.

Bon courage ! 加油

(Go to YouTube to follow the whole series of lectures by Prof Gross; or check Harvard Online website to download the videos and lecture notes.)

Ref:

https://tomcircle.wordpress.com/2013/05/10/harvard-abstract-algebra-video/

Note: Spend one of two nights per week of 1.5 hours to attend the lectures. You can complete the whole course in 6 months.

# Online Study Guide : Abstract Algebra

http://www.math.niu.edu/%7Ebeachy/abstract_algebra/study_guide/contents.html

The Study Notes on 600 problems and solutions:

http://www.math.niu.edu/~beachy/abstract_algebra/guide/contents.html

# Groups Lecture Notes (Cambridge Math Tripos)

# Solution Ecole Polytechnique -Ecole Normales Superieures Concours 2013

We shall walk through the problem at little steps and day-by-day, not so much interest in the final solution* per se *, but with a higher aim to revise the modern algebra lessons along the way.

The French professors who designed this problem had done beautifully using all the concepts learned in the 2-year Classe Préparatoire (or **Prépa,** equivalent to Bachelor degree in Math & Science) – it is like an orchestra composer who pieces together all instruments to play a beautiful symphony – the catch is that the student must have a good grasp of all algebra topics.

**SOLUTION**

I – **Operators on the functions with finite support**

1. a. Prove that **V** is a vector subspace of .

**Proof:**

Recall the definitions:

= v.s.{}

Support = **supp(ƒ)** = {}

**V ** = {f | supp(f) is a finite set}.

To prove V a vector subspace of ,

1) **V must be non-empty**?

V contains null function, so not empty subset of

2) **closed under vector addition**?

supp (f+g) supp(f) supp (g)

3) **closed under scalar multiplication**?

supp() = supp(f) for

Given ƒ ∈ , E(ƒ) ∈

by E(ƒ)(κ)= ƒ(κ+1), κ ∈

1. b. Show that E∈ L( ) and V is stable by E.

E by definition is an operator of shift, hence a linear transformation, thus

E∈ L( ).

——

**[Solutions for XLC paper]:**

# Translated Ecole Polytechnique & Ecole Normales Superieures Concours 2013

Math Paper A (XLC)

Duration: 4 hours

Use of calculator disallowed.

We propose to study the algebras of the remarkable endomorphisms of vector spaces of infinite dimension.

**Preamble**

An root of unity is called **primitive** if it generates the group of roots of unity.

In this problem, all vector spaces are over the base field of complex numbers field .

If **ε** is a vector space, the algebra of the endomorphisms of ε is denoted by **L(ε)**, and the group of the automorphisms of ε is denoted by **GL(ε)**.

denotes the identity mapping of ε.

If u ∈ L(ε), denotes the sub-algebra of L(ε) of the Polynomials in u.

denotes the vector space of the functions of to .

If ƒ is the function of to , **supp(ƒ)** denotes the set of κ ∈ such that ƒ(κ) ≠0

We call this set the **support** of ƒ.

Throughout the problem, **V **denotes the set of functions of to of which the support is a finite set.

I – **Operators on the functions with finite support**

1. a. Prove that **V** is a vector subspace of .

Given ƒ ∈ ,

we define E(ƒ) ∈

by

E(ƒ)(κ)= ƒ(κ+1), κ ∈

1. b. Show that E∈ L( ) and V is stable by E.

In the following, E denotes uniquely the endomophism of V induced.

2. Show that E ∈ GL(V).

3. For , we define by:

3.a. Prove that the family is the base of V.

3.b. Calculate E().

Let ,

we define the respective linear mappings

by:

and

4. Prove that

if and only if for all

**In the remaining of Section I (but not in the following Sections), we asume the conditions in question 4 are verified.**

5. Prove that

if and only if

6.a. Prove that for the vector space generated by has finite dimension.

6.b. Deduce that a vector subspace non-reduced to {0} of V, stable by H, contains at least one of the .

**In the remaining of Section I (but not in the following Sections), we asume the conditions in question 5 are verified and **

7.a. Prove that

7.b. Prove that E and F are not of finite order in the group GL(V).

7.c. Calculate the kernel of H and prove that

8. denotes the polynomials with complex number coefficients in one indeterminate X.

8.a. Prove that is isomorphic (as Algebra) to .

8.b. Prove that is isomorphic (as Algebra) to .

8.c. Prove that is isomorphic (as Algebra) to .

**II – Interlude**

In all the rest of the problem, we fix an odd interger and** q **a primitive root of unity.

9. Prove that is a primitive root of unity.

Let

10. Consider the element is :

10.a. Calculate .

Prove that

10.b. Let **b** root of **a**.

Calacute the eigenvectors of and the associated eigenvalues in function of b, q and .

Let’s define a linear mapping by

, and

define r and p respectively the residue and the quotient of the euclidian division of i by ℓ; ie:

11. Prove that is a projector of image .

**III – Quantum Operators**

12. Prove that

if and only if

**In the following problem, we asume the conditions in question 12 are verified and **

13. Prove that .

14. Prove that

if and only if

**In the following problem, we asume the conditions in question 14 are verified.**

15.a. Prove that are periodic over , of periods dividing .

15.b. Prove that the period of .

15.c. Prove that the period of is also equal to .

16. Let with being inverse of H.

16.a. Prove that

.

16.b. For , prove that is an eigenvector of C.

16.c. Deduce that C is a homothety of v of which we calculate the ratio of in function of .

16.d. Let’s fix . Prove that the mapping

is a bijection of onto .

16.e. Let’s fix . Prove that the mapping

is a surjection of onto but not a bijection.

**IV – Modular Quantum Operators**

Let like in the Section II. We say an element of L(V) is **compatible** with if

17.a. Prove that if is commutative with ,then is compatible with .

17.b. Prove that are compatible with .

Let the set of endomorphisms which are compatible with .

18. Prove that is a sub-algebra of L(V).

19. Prove that .

20.a. Show that there exists an unique morphism of algebras such that:

20.b. Prove that is contained in the kernel of if and only if the image of is a vector subspace of v generated by the vectors where is the euclidian division of .

21. Let’s study in this question .

21.a. Determine .

21.b. Deduce .

21.c. Calculate the dimension of the vector subspace of

21.d. Calculate the eigenvectors of

22. Let W a non zero sub-space of stable by .

22.a. Show that W contains at least one of the vectors .

22.b. What do you say if W is in addition stable by ?

23. Give a necessary and sufficient condition on in order for the operator to be nilpotent.

—End—

**[Source]:**

http://www.ilemaths.net/maths_p-concours-polytechnique-mp-2013-01.php

# Harvard Abstract Algebra Video

The excellent lecture videos of “**Introduction to Abstract Algebra**“, taught by Prof Benedict Gross at Harvard, can be downloaded here:

http://www.extension.harvard.edu/openlearning/math222/

I met Prof Benedict Gross in Singapore last year at the NUS Public Lecture. I thanked him for these Abstract Algebra videos recorded many years earlier which helped me to understand and follow this advanced public lecture on “Elliptic Curve”. He was thrilled that remotely he could influence an unknown student through his Internet lectures.

Ref Books : Two Masterpieces on Abstract Algebra:

1) Written by MIT Prof, used by Harvard Prof Gross in this Online Course

Algebra (2nd Edition) (Featured Titles for Abstract Algebra)

2) Classical Approach:

Abstract Algebra, 3rd Edition

# Think Abstractness

Why some students can learn abstract Math easily while most can’t ?

Education Psychologists help to reduce abstraction level when learning Abstract Math Concepts.

There are 3 abstract levels:

**1st Level** : Quality of the relationship between the object of thought and the students:

The opposite of abstract is concrete.

Some students can relate the abstract math objects to concrete familiar concepts, the closer the relationship the more concrete the objects are.

eg.

Relate abstract ‘**Ring**‘ to familiar concrete object ‘**Integer Z**‘.

**2nd Level** : Process conception and Object conception:

The mental process that leads from process conception to object conception is called “* Reflective Abstraction*“. (Piaget).

eg. Quotient Group = G/H = {Hg = gH | g ∈ G}

**Process Conception** (Canonical procedure ) : take all elements from H, multiply them on the right with some element from G. Similarly for multiplication on the left.

The above “Process” leads to below …

**Object Conception**: Partition called Quotient Group. Example: my favorite “Object Conception” for Group Kernel with the “King of fruits” Durian: https://tomcircle.wordpress.com/2013/04/11/from-durian-to-group-theory/

**3rd Level** : Degree of Complexity of the concept:

eg. Group of prime p, called p-iadic group is more complex than any group.

Music is abstract par excellence, hence a good training for kids to study later abstract subjects in Advanced Physics or Advanced Mathematics. Great scientists like Albert Einstein, Niels Bohr and great Mathematicians played quite well some musical instrument (Violin for Einstein & Bohr).

# Abstract Algebra compulsory

*Maths, Science, Engineering, IT, Finance*:

Visual Group Theory (MAA Classroom Resource Materials) (MAA Problem Book Series)

# Abstract Math discomforts

3 Wide Discomforts For Abstract Math Students

1. Group : Coset, Quotient group, morphism…

2. Limit ε-δ: Cauchy

3. Bourbaki Sets: Function f: A-> B is subset of Cartesian Product AxB.

Students should learn from their historical **genesis** rather than the formal abstract definitions

<a href=”http://http://en.wikipedia.org/wiki/Wu_Wenjun“>Wu Wenjun (吳文俊) on Learning Abstract Math

“…*It is more important to understand the ‘Principles’ 原理 behind, à la Physics (eg. Newton’s 3 Laws of Motion), and not blinded by its abstract ‘Axioms’ 公理*.”

**Prof I.Herstein http://en.wikipedia.org/wiki/Israel_Nathan_Herstein**

“… *Seeing Abstract Math for the first time, there seems to be a common feeling of being adrift, of not having something solid to hang on to.*”

“*Do not be discouraged. Stick with it! The best road is to look at examples. Try to understand what a given concept says, most importantly, look at particular, concrete examples of the concept.*”

“*Abstract Math plays a dual role: that of unifying link between disparate parts of math and that of a research subject with a highly active life of its own. It plays an ever more important role in physics, chemistry, and computer science, etc*.”

See also: How to think abstractness

# 4-level MathThinking

4 Levels:

L1. S&T (See & Touch) Concrete: 1 apple, 2 oranges…

e.g. Math Modeling: visualise the problem [*Primary School*]

L2. S~T (See, no Touch but can guess):

e.g. Guess x,y for 2x+3y=8 ? [*Secondary Schoo*l]

e.g. Chimpanzees can guess where you hide the banana.

L3. ~S~T&I (no See, no Touch but Imagine):

e.g. Complex i = [*Junior College*].

L4. ~S~T~I (no See, no Touch, no imagine)

e.g. Abstract Math: Galois Group, ε-δ Analysis, Ring, Field, etc. [*University*]