# 6.13 Field Homomorphisms

Field homomorphisms are an important class of homomorphisms in GAP (see chapter Homomorphisms).

A field homomorphism phi is a mapping that maps each element of a field F, called the source of phi, to an element of another field G, called the range of phi, such that for each pair x,y in F we have (x+y)^phi = x^phi + y^phi and (xy)^phi = x^phi y^phi. We also require that phi maps the one of F to the one of G (that phi maps the zero of F to the zero of G is implied by the above relations).

An Example of a field homomorphism is the Frobinius automorphism of a finite field (see FrobeniusAutomorphism). Look under field homomorphisms in the index for a list of all available field homomorphisms.

Since field homomorphisms are just a special case of homomorphisms, all functions described in chapter Homomorphisms are applicable to all field homomorphisms, e.g., the function to test if a homomorphism is a an automorphism (see IsAutomorphism). More general, since field homomorphisms are just a special case of mappings all functions described in chapter Mappings are also applicable, e.g., the function to compute the image of an element under a homomorphism (see Image).

The following sections describe the functions that test whether a mapping is a field homomorphism (see IsFieldHomomorphism), compute the kernel of a field homomorphism (see KernelFieldHomomorphism), and how the general mapping functions are implemented for field homomorphisms.

GAP 3.4.4
April 1997