Fielddenoting

Fielddenoting is a term used in some mathematical contexts to refer to a construction or notion that assigns to a given object a field which serves as its denoting or defining field. It is not a standardized term across the literature, and its precise meaning can vary by author or subfield. The common idea is to capture a canonical field that is minimal or natural for describing, realizing, or comparing the object in question.

In practice, fielddenoting often centers on two related ideas. First, the field of definition: the smallest field over which the object can be described by equations or coordinates. Second, a coordinate or realization field: the field generated by the data needed to realize the object, such as coefficients of defining equations or coordinates of a representation. In some settings, the fielddenoting construction is required to satisfy a universal or functorial property, so that any other realization factors through the designated field.

Examples include:

- The field of definition of an algebraic variety: the minimal subfield of a fixed algebraically closed field over which the defining equations can be written.

- The splitting field of a polynomial: the smallest field over which all roots lie, often considered a canonical field associated to the polynomial.

- The field of definition or moduli for a geometric object with additional structure, where distinctions between base fields and fields of moduli are studied.

Relation to related concepts includes field of moduli, base field, and definability. The term remains informal in many contexts and should be interpreted with attention to the specific definitions used in a given work.