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If p is prime, prove the group (Zp*,x) has exactly one element of order 2.

  1. Nov 26, 2011 #1

    ae1709

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    Hi. I need to: prove the group (Zp*,x) has exactly one element of order 2. Here, p is prime and (Zp*,x) is the set {1, 2,....., p-1} under multiplication modulo p. Any help would be much appreciated!
     
    ae1709, Nov 26, 2011
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  3. Nov 26, 2011 #2

    Office_Shredder

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    I think the easier half is: which element has order two?

    It might help if you write down the condition of what it means to have order 2
     
    Office_Shredder, Nov 26, 2011
  4. Nov 26, 2011 #3

    micromass

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    Something that might help you is knowing that [itex]\mathbb{Z}_p^*[/itex] is cyclic. Do you know this already??
     
    micromass, Nov 26, 2011
  5. Nov 26, 2011 #4

    micromass

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    If you do not know the previous. Then you might want to think about the polynomial [itex]X^2-1[/itex] in [itex]\mathbb{Z}_p[X][/itex]. What do you know about its roots??
     
    micromass, Nov 26, 2011
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