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## Homework Statement

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.

## Homework Equations

## 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}= 1

^{2}, it follows that 0 = p

^{2}- 2p, and hence p = 2, a contradiction.

What can i do to find a correct way of proving this?