fieldsI

In mathematics, fieldsI denotes a family {F_i}_{i∈I} of fields indexed by a set I. The index set I may be finite or infinite. When treated as a diagram, fieldsI can be described by a functor from a small index category I to the category of fields, encoding a structured collection of fields with maps between them.

Basic constructions with fieldsI include the Cartesian product ∏_{i∈I} F_i, which carries a natural ring structure by coordinatewise addition and multiplication. The product is generally not a field unless I is a singleton or all but one component coincide with a common base field. The directed limit or colimit of a directed system of fields contained in a common overfield can also be a field under suitable conditions.

Examples: If I = {1,2} with F_1 = Q and F_2 = R, then fieldsI consists of the two fields Q and R. If I is a singleton, fieldsI reduces to that single field. In a more categorical setting, fieldsI may be viewed as a functor I → Fields, describing a family of field extensions parameterized by I.

Applications: The notion appears in algebraic geometry, number theory, and model theory to model fiberwise or parameterized algebraic data, to discuss product and limit constructions of fields, and to study families of field extensions and their interrelations.

Notes: The exact meaning of fieldsI depends on context, and the term is not universally fixed. Consult the surrounding text to determine the intended structure. See also: field, product of fields, directed limit, ultraproduct.