Friendship Theorem

In a party of n people, in which any 2 persons know each other, then there is a person who knows everybody.

Field: Galois, Dedekind

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)

http://www.google.com.sg/imgres?imgurl=http://info.makepolo.com/uploadfile/2012/0723/20120723100653765.jpg&imgrefurl=http://info.makepolo.com/htmls/6/69/2669.html&h=400&w=533&sz=44&tbnid=ExodLfHv3cQjHM:&tbnh=91&tbnw=121&zoom=1&usg=__hsZaBecpPNdvTvguQbaQftCsXgo=&docid=qXMWtmo8A-vXEM&hl=en&sa=X&ei=f2NaUciqKYrOrQeT2YHgDw&sqi=2&ved=0CEsQ9QEwAg&dur=591

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.

Gauss saved by French Lady

Gauss and Sophie Germaine

Gauss was from Brunswick, now Germany. His king and academic sponsor was killed by Napoleon Army. Gauss had a French pen-friend "Mr. Brun" who was the famous French lady mathematician Sophie Germaine in disguise. She asked Napolean's general not to kill Gauss. He spent remaining life in the University of Göttingen, which produced Klein, Hilbert, Riemann... Göttingen replaced Paris to be the World Center of Math until Hitler destroyed it.

Gauss rejected by French

Gauss, at 21, wrote the world's 1st Number Theory Book “Disquisitiones Arithmeticae” on his famous inventions (Modulus arithmetic, Quadratic Reciprocity Theorem, 17-sided polygon). It was rejected by Paris Academy of Science. The French rejection caused him a life-long reluctance, like Newton, to publish his works (e.g. Non-Euclidean Geometry). He also erased all the proofing steps, only showed the end results.

Vedic (Equation)

Bi-quadratic Equation by Vedic Math

Solve
$latex (x+7)^4+(x+5)^4=706$
Let y = x + 6 = average of (x+5, x+7)
$latex (y+1)^4+(y-1)^4=706$
Cancel terms y, y³:
$latex 2y^4+12y^2+2=706$
$latex y^4+6y^2-352=0$
$latex y^2=16$ or
$latex y^2=-22$
y= ±4 or ±$latex \sqrt{-22}$
y=x+6
x=-2, -10, ± $latex \sqrt{-22}-6$

Vedic (GCD Polynomials)

G.C.D Polynomials by Vedic Math

Find G.C.D of P(x) & Q(x):

P(x) = 4x³ +13x²+19x+4
Q(x) = 2x³+5x²+5x -4

Vedic method:
1. Eliminate 4x³ in P(x):
P - 2Q = 3x² +9x+12

/3 => P-2Q = (x²+3x+4)

2. Q+P = 6x³+18x²+24x

/(6x) => Q+P = (x²+3x+4)

3. G.C.D. = (x²+3x+4)

P= (x² +3x+4).(ax+b) = 4x³ +13x²+19x+4
=> a=4, b=1
Similarly,
Q= (x² +3x+4).(2x+1) = 2x³+5x²+5x -4

Quotations

1. Issac Newton: Hypotheses non fingo (I frame no hypotheses)

2. Gauss: Pauca sed matura (Few but ripe)

3. Descartes: Bene vixit qui bene latuit (he has lived well who has hidden well.)