Suppose that we want to construct a field of order 81 (= 34). Then we should look for an irreducible polynomial f of degree 4 in Z/3Z[X].
According to the theory, f is a divisor of the polynomial X81 - X. We first divide out the roots belonging to the subfield of order 9:
This polynomial will factor into 18 irreducible polynomials of degree 4. We find one by trial and error: Creating a degree 4 polynomial and checking that it is relatively prime with X9 - X. The 18 choices for f that may arise are:
| X4 - X2 - 1 | X4 + X2 - X + 1 |
| X4 - X3 + X2 + 1 | X4 + X3 - X + 1 |
| X4 + X3 + X2 - X - 1 | X4 + X2 - 1 |
| X4 - X3 - 1 | X4 + X - 1 |
| X4 + X3 - 1 | X4 - X3 + X + 1 |
| X4 - X3 + X2 + X - 1 | X4 + X2 + X + 1 |
| X4 - X3 - X2 + X - 1 | X4 - X3 + X2 - X + 1 |
| X4 + X3 - X2 - X - 1 | X4 + X3 + X2 + X + 1 |
| X4 - X - 1 | X4 + X3 + X2 + 1 |