Example

We study fields of order 9. The polynomial X⁹ - X Z/3Z[X] factors as follows

X(X + 1)(X + 2)(X² + X + 2)(X² + 2X + 2)(X² + 1).

The first three factors are linear, and correspond to the zeros of the polynomial X⁹ - X in Z/3Z. The other three irreducible factors are quadratic. Each of them can be used to define a field of order 9.

In the next theorem we shall see that they all lead to the same field (up to isomorphism, that is, Z/3Z[X]/(X² + X + 2), Z/3Z[X]/(X² + 2X + 2), and Z/3Z[X]/(X² + 1) are isomorphic to each other).