A number of basic fields are provided, such as the rationals, finite fields, the real field, etc.
Various generic field constructions can then be made recursively on top of these basic fields. For example, fraction fields, residue fields, function fields, etc.
From the point of view of the system, all fields are rings and whether an object is a ring/field or an element thereof can be determined at the type level. There are abstract types for all field and for all field element types.
The field hierarchy can be extended by implementing new fields to follow one or more field interfaces, including the interface that all fields must follow. Once an interface is satisfied, all the corresponding generic functionality will work over the new field.
Implementations of new fields can either be generic or can be specialised implementations provided by, for example, a C library.