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Finite Fields

  1. Jan 20, 2008 #1
    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. jcsd
  3. Jan 20, 2008 #2
    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.
     
  4. Jan 20, 2008 #3
    actually it's p^n for some positive integer n, but the same thing...
     
  5. Jan 20, 2008 #4

    Dick

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    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.
     
  6. Jan 20, 2008 #5
    but why isn't it possible for an element besides 1, -1, to have order 2 so that its inverse is itself?
     
  7. Jan 20, 2008 #6

    Dick

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    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.
     
  8. Jan 20, 2008 #7
    awesome :) just what i needed. thnx
     
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