nLab Field

Redirected from "category of fields".
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Definition

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.

Properties

Subcategories

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))

References

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