Let p be a odd prime, If ord p (a) = h and h is even, then a^(h/2)= -1 mod p

Question

Determine is, in general, true or false. Recall that a
universal statement is true if it is true for all possible cases while it is false if there is even one
counterexample. Be prepared to prove that your answer is correct by supplying a proof or
counterexample, whichever is appropriate

Let p be a odd prime, If ord p (a) = h and h is even, then a^(h/2)≡ -1 mod p


solution:

ord p (a) = h
a^(h)≡ 1 mod p
a^(h) -1 =pk where k is an integer.
since h is even
(a^(h/2) -1)(a^(h/2) +1)=pk
k=[(a^(h/2) -1)(a^(h/2) +1)]/p

since ord p (a) = h
(a^(h/2) -1) is not divisible by p
thus,

(a^(h/2) +1) is divisible by p

(a^(h/2) +1)≡0 mod p

a^(h/2)≡ -1 mod p

Could you check it for me please is it correct or not?

Answer 1

Your solution looks excellent!