Working with finite fields of form Z_p

[Syrus]

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 = 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?

 

[jgens]

In a finite field, the group of units is cyclic. This gives you a quick way of proving the result.

 

[Dick]

It's probably even easier if you simply observe that if i^2=k then (-i)^2=k. But since your first proof is so bogus, you probably really don't understand what Z_p is. Try doing a warmup and figuring out what elements of Z_3, Z_5 and Z_7 are squares and which aren't.

 

[SteveL27]

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.

Are you allowed to divide by p in Z_p?

For example suppose we're in Z₅. Now

5 * 2 = 10 = 0 (mod 5)

and 5 * 3 = 15 = 0 (mod 5).

So since 5*2 = 5*3 we conclude that 2 = 3.

What's wrong with that proof?

 

[Syrus]

Yes, I suppose that's silly... dividing by p (which is congruent to zero). I have done what dick suggested and discovered that for Z_3, 2 is not a square. For Z_5, neither 2 nor three are squares. I am trying to work off this for now but can't seem to generalize these cases.

 

[Dick]

Yes, I suppose that's silly... dividing by p (which is congruent to zero). I have done what dick suggested and discovered that for Z_3, 2 is not a square. For Z_5, neither 2 nor three are squares. I am trying to work off this for now but can't seem to generalize these cases.

Ok, I think you got what Z_p means. Now use counting logic. If every element of Z_p were a square then for every element z of Z_p there is an element k such that k^2=z. If 1^2=1 and (-1)^2=(p-1)^2=1 do you have enough elements of Z_p to give each element a square root?