fieldereither

Fieldereither is a theoretical construct used in abstract algebra and logic to formalize a disjunctive choice between two standard subfield constructions of a common base field within a fixed ambient field. The term combines field theory with the logical connective or, signaling a selection between two alternative field-building operations.

Definition and construction

Given a base field F and two field extensions K and L with F embedded in both, a fieldereither FE(K,L;s) is a subfield of a chosen ambient field E determined by a selector function s that maps the pair (K,L) to either the value intersect or compositum. If s(K,L) = intersect, then FE(K,L;intersect) equals the intersection K ∩ L. If s(K,L) = compositum, then FE(K,L;compositum) equals the compositum KL, the smallest subfield containing both K and L. In this sense, a fieldereither serves as a formal device for expressing a conditional construction within a single model.

Examples

Let F = Q, K = Q(√2), and L = Q(√3). A fieldereither FE(K,L;intersect) yields Q, while FE(K,L;compositum) yields Q(√2,√3). The concept highlights how different natural field constructions can be encapsulated under a single notation by varying the selector.

Usage and notes

Fieldereither is not standard in mainstream textbooks and is mainly encountered in expository writings or discussions about conditional field constructions in model theory and algebra. It does not define a new algebraic object by itself, but rather a convenient way to discuss alternative subfield formations within a unified framework.

See also

Field, Field extension, Intersect, Compositum, Lattice of subfields.