Three mysterious Math Objects:

- Pi,
- Complex Numbers,
- Prime Numbers

are hiding in circle.

took 300 Years to be accepted.

1. 17AD – Cardano: (stole it from other) for cubic equation.

2. 18AD – Lebniz: **i is like “Holy Spirit” between ‘there’ and ‘not there’.**

3. 19AD – Gauss: i as Geometry

Point (a,b) in plane RxR :

z=a+ib

a on x-axis, b on i-axis

4. 20AD – Hamilton:

i as ordered-pair (a,b) with unique operations:

(a, b)+(c,d)=(a+c, b+d)

(a, b).(c,d)=(ac-bd, bc+ad)

Emile Artin’s very unique book “Galois Theory” (1971) on “**Finite Field Extension**” interpreted by **Vector Space**.

Let H a Field with subfield G

F is G’s subfield:

H ⊃ G ⊃ F

Example:

Let

F = Q = Rational Field

G = Q(√2) = Larger Extended Field Q with irrational root √2

H = Q(√2, √3) = Largest Extended Field Q with irrational roots (√2 & 3)

{1, √2} forms basis of Q(√2) over **Q**

{1, √3} basis of Q(√2, √3) over **Q(√2)**

**[**since** **√3 ≠ p+ q√2 , ∀p,q ∈ Q]

=> {1,√2, √3, √6} basis of Q(√2, √3) over **Q**

=> Q(√2, √3) is a 4-dimensional Vector Space over **Q**.

**Q(√2) **≌ **Q[x] / {x****² – 2}**

Read as:

**Why below are not Fields** ?

R[x] / {x³ + 1}

R[x] / {x^4 + 1}

**C**[x] / {x**²** + 1}

*Hint: they are not irreducible in that particular Field, not a Principal Ideal.*

Note: **C[x] **the Polynomial Ring with coefficients in the Field C

Ask: NZQRC…X?

(*Nine Zulu Queens Rule China* …)

Is there a number system X beyond Complex C?

We know that Z is extended of N

because of solving equation like :

y+2=0

=> y= -2 ∈Z

Similarly Q extends Z:

3y-2 =0

=> y= 2/3 ∈ Q

R extends Q:

y² = 2

=> y = √2 ∈ R

C extends R:

y² = -1

=> y = √-1 ∈ C

What about something extends C ?

Answer: No such number system !

Why ?

Let the Polynomial equation

P(x) = xⁿ +…+ dx³+ cx² + bx + a

P(x) has n solutions in C (by Gauss Fundamental Law of Algebra): z0, z1, z2, …zn-1

P(x) can be factorized as:

P(x) = (x-z0).(x-z1).(x-z2)….(x -zn-1)

P(x)=0 has all n complex solutions still in C

=> **closed** in **C** or **Complete** in **C**

=> no need to extend C like the previous number systems (NZQR)