4.5 Domain

Domain( list )

Domain returns a domain that contains all the elements in list and that knows how to make the ring, field, group, or vector space that contains those elements.

Note that the domain returned by Domain need in general not be a ring, field, group, or vector space itself. For example if passed a list of elements of finite fields Domain will return the domain FiniteFieldElements. This domain contains all finite field elements, no matter of which characteristic. This domain has a function FiniteFieldElementsOps.Field that knows how to make a finite field that contains the elements in list. This function knows that all elements must have the same characteristic for them to lie in a common field.

    gap> D := Domain( [ Z(4), Z(8) ] );
    FiniteFieldElements
    gap> IsField( D );
    false
    gap> D.operations.Field( [ Z(4), Z(8) ] );
    GF(2^6) 

Domain is the only function in the whole GAP library that knows about the various types of elements. For example, when Norm is confronted by a field element z, it does not know what to do with it. So it calls F := DefaultField( [ z ] ) to get a field in which z lies, because this field (more precisely F.operations.Norm) will know better. However, DefaultField also does not know what to do with z. So it calls D := Domain( [ z ] ) to get a domain in which z lies, because it (more precisely D.operations.DefaultField) will know how to make a default field in which z lies.

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