1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. 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?