Prove that a^t = -1 (mod p^k) for

[xax]

Prove that a^t = -1 (mod p^k) for...

p<>2, prime and ord p^k (a) = 2t.

 

[robert Ihnot]

I am not certain about the meaning of p<>2, but if the order of a is 2t, then a^t==-1 Mod p^k...Because there are only two elements of order 2, and phi(2) =1, which tells us only one element belongs to 2. That is, if x^2==1 Mod P^k, then (x-1)(x+1) == 0 Mod P^k. So in an integral domain, one of the terms under discussion (a^t-1) or (a^t+1) will be equal to zero, Mod p^k.

 

[xax]

Thanks robert for your help, but how can I say that only a^t+1 = 0 mod p^k and it's not possible a^t-1=0 mod p^k?
Edit: p<>2 means p can't be 2.

 

[robert Ihnot]

Because the order is even. If a^t-1 = 0 Mod p^k, then the order is odd.

 

[xax]

got it robert, thank you.