Find all c epsilon Z5 for which Z5 xx3 2x c is a field Exp

Find all c epsilon Z_5 for which Z_5 [x]/(x^3 + 2x + c) is a field. Explain your reasoning.

Solution

For Z5(x)/(x^3 + 2x +c) to be field p(x)= x^3 + 2x +c

should be irreducible to Z5(x) as p(1) p(2) p(3) p(4) p(5) value corresponds to 3 +c,12 +c,33+c,72+c,135+c respectively.

To be irreducible c has to be positive as cycle should not form to be irreducible.

So c>=0 for this to be field.

Thank you