Let p be an odd prime. Then Char(Z_p) is nonzero.
Prove: Not every element of Z_p is the square of some element in Z_p.
The Attempt at a Solution
I first did this, but i was informed by a peer that it was incorrect because I was treating the congruency as an equality:
Suppose not. Then every element of Z_p is the square of some element in Z_p. Take 1. Since in mod p: 1 = (p-1)2 = 12, it follows that 0 = p2 - 2p, and hence p = 2, a contradiction.
What can i do to find a correct way of proving this?