Understanding Field Theory: Finite Fields and Clock Arithmetic Explained

[2710]

Hi,

Can someone briefly explain Field theory to me?

Ie, in this book, it says that 1+1 = 0 when field : F (subscript 2)

And you can create a finite field for any prime number p. I don't understand this lol.

I also got this table:

+ 0 1 2 3
0 0 1 2 3
1 1 0 3 2
2 2 3 0 1
3 3 2 1 0

and Multiplication:

x 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 3 1
3 0 3 1 2

F(Subscript 4)

How do I go about finding these numbers? 3x3 is 9 as far as I am concerned, unless he is using clock arithmetic, in which case 3x3 would be 1 on a 4 Clock... its 2 here T___T

Help appreciated :D

 

[guildmage]

That field is the Galois field of order 4, denoted $$GF(2^2)$$ or $$F_{2^2}$$. The operations there are addition modulo 4 and multiplication modulo 4 (I think this is what you meant by clock arithmetic). We can form finite fields $$GF(p^r)$$ or $$F_{p^r}$$ of order $$p^r$$ where $$p$$ is a prime.

 

[2710]

But they don't. If you look closely, 3+3 = 6 which is 2 modulo 4. But its 0 on my table. This isn't a modulo 4 table is it?

Thanks

 

[DrGreg]

If you look closely, 3+3 = 6 which is 2 modulo 4. But its 0 on my table. This isn't a modulo 4 table is it?

No, you've got the wrong table. The modulo 4 tables are

+ 0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3 3 0 1 2

× 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1

(i.e. "clock arithmetic") Denoted Z₄. This is not a field because 2 does not have a multiplicative inverse; there in no solution to 2x = 1.

Z_n is a field if and only if n is a prime number.

 

[2710]

no! I haven't got the wrong table. I know that my tables aren't modulo 4 tables that's what I am saying. I don't care about modulo 4 tables. I want to know what MY tables are. In the book they say its a F4 table, but I don't understand what this means.

 

[HallsofIvy]

guildmage did not say anything about "modulo arithmetic". The only fields using "modulo arithmetic" are Z_p for p prime, as you said. If i is not prime, say i= mn, the Z_i is not a field because m*n= i= 0 (mod i). A field cannot have "zero-divisors". guildmage specifically referred to the tables you give as the "Galois Field of order 4" and that is NOT using "modulo" arithmetic.

 

[2710]

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...

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?

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

EDIT: Also, how do you work out the elements in a field? F2 only has 0 and 1, why is this?

Thanks :D

 

[HallsofIvy]

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"!

 

[2710]

Oh, lol, sorry for getting you worked up :P

It is actually Galois' Field order 4

So, every F(2^(N)) Galois Field has the same table as F (subscript 2)?

And yeh, I am just reading this book, I might take a full fledged course in the future :D

Thanks, I guess Ill just have to take for granted that 3+3 = 0 (in the galois 4 table field) :D