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.
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.