Example

In order to construct a field of 9 elements, we have to find an irreducible polynomial of degree 2 in Z/3Z[X].

The polynomials of degree 2 with leading coefficent 1 are:

X2 + X + 1		X2 + 1
X2 - X + 1		X2 - 1
X2 + X - 1		X2 + X
X2 - X - 1		X2 - X
X2

The irreducible polynomials among them are X² + X + 1, X² - X - 1, and X² + 1.

So, we can construct a field of 9 elements by taking the residue class ring S = Z/3Z[X]/(d) where d = X² + 1.

One of the special properties of finite fields is their uniqueness. For example, had we taken one of the other two irreducible polynomials of degree 2, we would essentially have obtained the same field.