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}}}}

{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}}
(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)。


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]

Modern Algebra (Abstract Algebra) Made Easy

UReddit Courses:

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

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.)


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.