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.

Eigenvector & Eigenvalue

1. Matrix (M): stretch & twist space
2. Vector (v): a distance along some direction
3. M.v = v' stretched & twisted by M

Some directions are special:-
a) v stretched but not twisted = Eigenvector;
b) The amount of stretch = constant = Eigenvalue (λ)

Let M the matrix, λ its eigenvalue,
v eigenvector.
By definition: M.v = λ.v
v = I.v (I identity matrix)
M.v = λI.v
(M - λI).v=0
As v is non-zero,
1. Determinant (M- λI) =0 => find λ
2. M.v = λ.v => find v

Note1: Why call Eigenvalue ?
From German: "Die dem Problem eigentuemlichen Werte"
= "The values belonging to this problem"
=> eigenWerte = EigenValue
Eigenvalue also called 'characteristic values' or 'autovalues'.
Eigen in English = Characteristic (but already used for Field).

Note2: Schrödinger Quantum equation's Eigenvalue = Maximum probability of electron presence at the orbit outside nucleus.

Note3: Excellent further explanation of the eigenvector and eigenvalue:

http://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html