Values that make a field (need answer checked)

I just need my answer checked for this particular problem.

The question is:
"For which values of a=1,2,3,4 is a field? Show your work.

My answer:
We need to know when the polynomial is irreducible in Z_5.
For this, we check to see if a is a perfect square in Z_5. The squares are:
0=0 1=1 2=4 3=4 4=1
So the values of a which arnt squares are 2 and 3, these being the values that give a field.

Correct?

Since is a quadratic polynomial, either has two linear factors or it's irreducible. If a polynomial has a linear factor then

So we check for every value of a, if and when

e.g.
. So is not irreducible.

I see!
So my method was wrong, but after rechecking, it seems 2 and 3 are indeed the correct values.

Your method was correct. You are both essentially doing the same thing - eliminating the values of for which the polynomial has a root.