Find all the monic irreducible polynomials of degree at most 3 over and .

Note that there are monic polynomials in of degree , as there are choices for each of coefficients.

Over :

All linear polynomials are irreducible; these are and .

The quadratic polynomials are , , , and which is irreducible since .

The cubic polynomials are , , , , which is irreducible since , which is irreducible since , , and .

Thus the irreducible polynomials of degree at most 3 over are , , , , and .

Over :

All linear polynomials are irreducible. These are , , and .

The quadratic polynomials are , which is irreducible since and , , , , , which is irreducible since and , , and which is irreducible since and .

The cubic polynomials are , , , , , , , which is irreducible since , which is irreducible since , , , which is irreducible since and , , , , which is irreducible since and , which is irreducible since and , , , which is irreducible since and , , , , which is irreducible since and , , , and which is irreducible since and .

Thus the irreducible polynomials of degree at most 3 over are , , , , , , , , , , , , and .