2710 said:
Ok, sorry for me being a noob, but I am only high school standard ¬__¬
Anyways, say you have Z16, why can't this be a field? Let 16=i, u say that m*n = i, so 2x8 = 16. And 2x8 = 0 (modulo 16), and then you say fields are not allowed to have zero-divisors. But I am not dividing by zero...
That's not what "zero-divisors" means. A "zero-divisor" is any element, a, such that for some b, ab= 0. In a field, every member, except 0, must have a multiplicative inverse. In your example of modulo 16, with x= 2, 2(0)= 0, 2(1)= 2, 2(3)= 6, 2(4)= 8, 2(5)= 10, 2(6)= 12, 2(7)= 14, 2(8)= 16= 0, 2(9)= 18= 2, 2(10)= 20= 4, 2(11)= 22= 6, 2(12)= 24= 8, 2(13)= 26= 10, 2(14)= 28= 12, 2(15)= 30= 14. There is NO y such that xy= 2y= 1 (mod 16). 2 does not have a multiplicative inverse so the "integers modulo 16" is NOT a field. It is a "ring" and that may be what you are thinking of.
Also, how about for F4? Coz I understand that the power of Primes are worked out differently. The book says something about characteristic of a F4 and F2 is 2. How do you work out the characteristic?
If by F4 you mean the integers modulo 4, they form a ring, not a field. Again, 2 is a "zero divisor" because 2(2)= 4= 0 (mod 4) and does not have a multiplicative inverse. The best way to "work out the characteristic" is to use the definition! The characteristic of a ring or field is defined as the number of times you add 1 (the multiplicative identity) to itself to get 0. The characteristic of F2 is 2 because 1+ 1= 0 in F2. In F4, 1+1= 2, 1+1+1= 3, 1+ 1+ 1+ 1= 4= 0 (mod 2) so the characteristic is 4, not 2.
That was assuming that you meant "integers modulo 4". If, instead, F4 is really F_{2^2}, the Galois field of order 4 that you had in your first post, then you can see from the "addition table" for that field that 1+ 1= 0 so the "characteristic" is 2.
I've never done field theory before, nor clock arithmetic, just read it in books. I want to know how the author of my book gets the table I wrote above.
Take 3+3 for F4 (Galois' thingy), on my table it gives 0. I just want to understand, plain and simple, how he gets this. I don't think anyone has answered my wquestion yet, coz you're all assuming I know the basics, which I dont, sorry :P
Do you understand that people take full year courses in "field theory"
after they have taken introductory courses in, say, discrete mathematics. We simply cannot give you a full course in field theory here.
EDIT: Also, how do you work out the elements in a field? F2 only has 0 and 1, why is this?
Thanks :D
Learn the
definitions! You ask, above, about "working out the characteristic" and, apparently have no idea what "characteristic" means (because how you "work it out" follows immediately from the definition). Now you are asking why F2 "only has 0 and 1". The answer is: because that is part of the
definition of "F2"!
