http://chinesepuzzles.org/nine-linked-rings/

Math Theory: Galois Field **GF(2)**

https://tomcircle.wordpress.com/2013/04/02/field-dedekind-web/

http://chinesepuzzles.org/nine-linked-rings/

Math Theory: Galois Field **GF(2)**

https://tomcircle.wordpress.com/2013/04/02/field-dedekind-web/

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

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.

————–

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

Example:

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

(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. *

Dedekind

(1831-1916)

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.