12-240/Fields' Further proof
12-240/Classnotes for Tuesday September 11
In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.
About: F(n)
F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)
If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"
Why all the numbers which are not prime numbers can not form a field F(n)?
Here is the proof.
If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.
IN defination of multiplication
- 0 1 2 3 .......... b.......n-1
0 0 0 0 0 ...........0........0
1 0 . . . ......................
2 0 . . . ......................
3 0 . . . ......................
4 0 . . . ......................
.. 0 . . . ......................
a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ( in this row, every element mod n)
....0.....................................
....0.....................................
....0.....................................
....0.....................................
(n-1) 0................................... see the (a+1)th row
There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")
So if F(n) is a field, then
there must exist k,m ∈N*, m<n
to meet the equation:
m*a=k*n+1
And we know that n=ab
So m*a=k*a*b+1 (a≠1)
Hence m=k*b+1/a
unless a=1
m will not exist, because m should be an integer. so F(n), when n is not a prime number, is not a field.
There is a large need for me to improve my format. Editing is welcomed.
PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang
