Finite field

A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields.

In the next sections, we will show how the general construction method outlined above works for small finite fields.

This may be deduced as follows from the results of the preceding section.

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field.

They form two orbits under the action of the Galois group. As the two factors are reciprocal to each other, a root and its (multiplicative) inverse do not belong to the same orbit.
They split into six orbits of six elements each under the action of the Galois group.

The fact that the Frobenius map is surjective implies that every finite field is perfect.

Number of monic irreducible polynomials of a given degree over a finite field