# 18.13 Field Functions for Finite Fields

Finite fields are, as the name already implies, fields. Thus all field functions are applicable to finite fields and their elements (see chapter Fields). This section gives further comments on the definitions and implementations of those functions for finite fields. All domain functions not mentioned here are not treated specially for finite fields.

`Field` and `DefaultField`

Both `Field` and `DefaultField` return the smallest finite field containing the arguments as an extension of the prime field.

`GaloisGroup`

The Galois group of a finite field F of size p^m over a subfield S of size q = p^n is a cyclic group of size m/n. It is generated by the Frobenius automorphism that takes every element of F to its q-th power. This automorphism of F leaves exactly the subfield S fixed.

`Conjugates`

According to the above theorem about the Galois group, each element of F has m/n conjugates, z, z^q, z^{q^2}, ..., z^{q^{m/n-1}}.

`Norm`

The norm is the product of the conjugates, i.e., z^{{p^m-1}/{p^n-1}}. Because we have Z(p^n) = Z(p^m)^{{p^m-1}/{p^n-1}}, it follows that Norm( GF(p^m)/GF(p^n), Z(p^m)^i ) = Z(p^n)^i.

GAP 3.4.4
April 1997