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

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}

Read as:

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
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


Bumpy-free CD/DVD

When you drive on a bumpy road, your CD music plays normally and smoothly while your body bumps up and down in your car?

Why CD disk can be bumpy-free ?
Thanks to Galois, the 19-year-old French Math genius and tragic victim of the X Concours. Galois invented the whole Abstract Algebra with two revolutionary ideas: Group and Field.
Galois Group explains why symmetry is a Math structure which plays the role of Quintic equation with no radical solutions.
Galois Field is used in the cryptography coding for computers, digital music in CD / DVD.
Galois Field Theorem: the only one in Field Theory
There is EXACTLY ONE such system whenever the number of element is a power of any prime number p.
Fp contains EXACTLY 1 sub-Field with pⁿ elements. (n ≥1)
For p prime, n ≥1
|Fp | = pⁿ unique
p=2 n =1
=> Galois Field = {0,1} with pⁿ = 2 elements

Galois Field {0,1}
1+1 = 0 = 0+0
1+0 = 1 = 0 + 1

Reed-Solomon codes for CD, DVD
(1960) using binary Galois Field {0,1}
Error correcting codes based on algebraic properties of Polynomials, whose coefficients are taken from a Galois Field.
255 bytes block = 223 + 32
223 = encode the signal
32 bytes = parity codes
When the car bumps and so the CD player’s laser beam on the CD disk, digital music code errors can be auto-corrected using Reed-Solomon code Algorithm, so you don’t hear any jerking disruption.

Field: Galois, Dedekind


Dedelind was the 1st person in the world to define Field:
“Any system of infinitely many real or complex numbers, which in itself is so ‘closed’ and complete, that +, – , *, / of any 2 numbers always produces a number of the same system.”

Heinrich Weber (1842-1913) gave the abstract definition of Field.

Field Characteristic

1. Field classification by Ernst Steinitz @ 1910
2. Given a Field, we start with the element that acts as 0, and repeatedly add the element that acts as 1.
3. If after p additions, we obtain 0 again, p must be prime number, and we say that the Field has characteristic p;
4. If we never get back to 0, the Field has characteristic 0. (e.g. Complex Field)

Example: GF(2) = {0,1|+} ; prime p = 2
1st + (start with 0):
0 + 1 = 1
2nd (=p) +:
1 + 1 = 0 => back to 0 again!
=> GF(2) characteristic p= 2

Galois Field GF(p)

1. For each prime p, there are infinitely many finite fields of characteristic p, known as Galois fields GF(p).

2. For each positive power of prime p, there is exactly one field.
(This is the only IMPORTANT Theorem need to know in Field Theory)
E.g. GF(2) = {0,1}

Math Game: Chinese 9-Ring Puzzle  (九连环 Jiu Lian Huan)

To solve Chinese ancient 9-Ring Puzzle (九连环) needs a “Vector Space V(9,K) over Field K”

finite Field K = Galois Field GF(2) = {0,1|+,*}
and 9-dimension Vector Space V(9,K):
V(0)=(0,0,0,0,0,0,0,0,0) ->
V(j) =(0,0,… 0,1,..0,0) ->
V(9)= (0,0,0,0,0,0,0,0,1)

From start V(0) to ending V(9) = 511 steps.