Integralfeld

Integralfeld is a fictional or hypothetical algebraic construct used in some theoretical discussions to merge the ideas of field theory and integral calculus. In this informal sense, an Integralfeld denotes a field F equipped with an integral operator that mirrors aspects of antiderivatives, defined on a chosen domain within F and satisfying algebraic compatibility with the field operations.

Definition. An Integralfeld (F, I) consists of a field F, a designated subspace Dom(I) ⊆ F called the domain of integration, and a linear map I: Dom(I) → F. The map I is required to satisfy linearity: I(a f + b g) = a I(f) + b I(g) for elements f, g in Dom(I) and scalars a, b from the base field. In addition, there is a fundamental theorem–like property: there exists a derivation d on F such that d(I(f)) = f for all f ∈ Dom(I). This structure is intended to resemble the relationship between integration and differentiation in calculus, within an algebraic frame.

Properties and discussion. An Integralfeld emphasizes linearity of the integral operator and its potential as a right inverse to a derivation on a subspace. The exact existence and form of I depend on extending the field or restricting to a suitable Dom(I); constants of integration arise as elements in the kernel of the derivation. In most standard algebraic settings, an integral operator does not live inside the same field unless the field is augmented.

Relation to other concepts. The idea of an Integralfeld is related to differential algebra, integral closures, and the study of integration operators in abstract algebra. It is not a standard or widely adopted notion in peer-reviewed mathematics and is more commonly found as a conceptual tool or in speculative discussions.

See also. Differential field, differential algebra, integration operator, Liouville theory.

References. The term is not widely standardized; for background, see foundational topics in differential algebra and algebraic approaches to integration.