nLab Field




FieldField is the category with fields as objects and field homomorphisms as morphisms. It is a full subcategory of CRing, the more general category of commutative rings, which is more often considered. This is due to FieldField not being a very well-behaved category.



FieldField is not connected as there are no field homomorphisms between fields of different characteristic. The connected component (full subcategory of FieldField) corresponding to characteristic pp (with p=0p=0 or pp prime) is denoted Field pField_p.

The field of rational numbers \mathbb{Q} is the initial object of Field 0Field_0 and the prime field 𝔽 p\mathbb{F}_p is the initial object of Field pField_p, but none are in FieldField, which has neither an initial nor terminal object. (Riehl 17, Examples 1.6.18. (vi))


Last revised on February 21, 2024 at 16:37:21. See the history of this page for a list of all contributions to it.