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.


One thought on “Field: Galois, Dedekind

  1. Pingback: Baguenaudier Chinese Rings 九连环 | Math Online Tom Circle

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