# Imaginary Numbers

FINAL EPISODE (13) – Riemann Complex Plane : 4 dimensions but viewed in 3 dimensions

Example: $\boxed { f (z) = z^2 + 1, z \in \mathbb {C}}$

Episode 1 – 9 : History

Episode 10: Complex function

Function : 1 input mapped to 1 output.

Multi-function : 1 input mapped to (n > 1 ) outputs.

# Mystery of i $\sqrt {-1} = i$ 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

i as ordered-pair (a,b) with unique operations:
(a, b)+(c,d)=(a+c, b+d)

# Artin Field Extension

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.

Isomorphism (≌)

Q(√2) Q[x] / {x² – 2}

Q(√2) isomorphic to the quotient of the Polynomial ring Q[x] modulo the Principal Ideal {x² – 2}
Q[x] the Polynomial Ring
{x² – 2} is the Principal Ideal
Complex Number (C)
C = R[x] / {x² + 1}
R[x] the Polynomial Ring with coefficients in the Field R
{x² + 1} is the Principal Ideal
Questions:
Since R[x] / {x² + 1} is the Field C

Why below are not Fields ?
R[x] / {x³ + 1}
R[x] / {x^4 + 1}
C[x] / {x² + 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 # Structure Beyond C

(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)