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

[ae1709]

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!

 

[Office_Shredder]

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

 

[micromass]

Something that might help you is knowing that $\mathbb{Z}_p^*$ is cyclic. Do you know this already??

 

[micromass]

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