I just need my answer checked for this particular problem.

The question is:

"For which values of a=1,2,3,4 is $\displaystyle Z_5[x]/<x^2+a>$ 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?