Definition
Category of fields
Fields and unit-preserving field homomorphisms.
The category of fields has fields as objects and unit-preserving ring homomorphisms between them as morphisms. All fields satisfy . Every morphism is injective: its kernel is an ideal in a field and cannot contain . Identities and composition are the usual functions.
Convention and comparison
A field extension supplies a morphism from its base field. An arbitrary linear map between the underlying vector spaces need not be a field morphism. In particular, there is no zero morphism between fields in this category.