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

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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!
 

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  • #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
 
  • #3
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Something that might help you is knowing that [itex]\mathbb{Z}_p^*[/itex] is cyclic. Do you know this already??
 
  • #4
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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??
 

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