Finite Fields

1. Jan 20, 2008

teleport

1. The problem statement, all variables and given/known data

Let F be a finite field. Show that the product of all non-zero elements of F is -1.

2. Relevant equations

An example of this is Wilson's Theorem.

3. The attempt at a solution

Let G be the multiplicative group of non-zero elements of F. Then G is cyclic. Let a be the generator of G. Here I get stuck. I thought that by just taking the product of each element represented by some power of the generator would be enough, but hey, it isn't. Not sure what to do now. Thnx for any help.

2. Jan 20, 2008

teleport

Can we just say that if F has characteristic p, then |G| = p - 1. Since G is cyclic then it isomorphic to Z*_p (which is also cyclic), and then use Wilson's Theorem and the isomorphism to conclude the product is -1.

3. Jan 20, 2008

teleport

actually it's p^n for some positive integer n, but the same thing...

4. Jan 20, 2008

Dick

There's a more direct way. Take the product of all the nonzero members a_1*a_2*...*a_n. For each a_i, a_i^(-1) is also in the list. Only +1 and -1 are their own inverses.

5. Jan 20, 2008

teleport

but why isn't it possible for an element besides 1, -1, to have order 2 so that its inverse is itself?

6. Jan 20, 2008

Dick

If a^2=1 then a satisfies a^2-1=0. Factor to (a-1)(a+1)=0. Since F is a field, one of those factors must be zero.

7. Jan 20, 2008

teleport

awesome :) just what i needed. thnx