field.cppm File Reference

The Rules of Harmony (Groups, Rings, and Fields). More...

#include <functional>
#include <type_traits>
import dedekind.algebra:universal;
import dedekind.sets;
import dedekind.order;
import dedekind.category;

Include dependency graph for field.cppm:

Namespaces
namespace	dedekind::sets

Concepts
concept	dedekind::algebra::HasFieldOperators
	Pure syntactic shape: T closes strictly under the full field operator surface +, binary -, unary -, *, /, and exposes the multiplicative unit T{1}.
concept	dedekind::algebra::IsField
	The set-indexed lift of the axiomatic category::IsField: a set object X whose carrier X::Domain is a field.
concept	dedekind::algebra::IsAlgebraicallyClosed
	The "Soul" of the Field: Every polynomial has a root in the species.

Functions
template<typename C> requires dedekind::category::IsField<C, std::plus<C>, std::multiplies<C>> && dedekind::category::IsClosedUnder<C, std::divides<>>
constexpr bool	dedekind::sets::admits (On< C >, FieldDivision, bool divisor_is_constant)
	Division by a nonzero constant is in a field's signature: multiplication by the inverse (category::IsField, the axiomatic claim; the carrier closes under /). Its free algebra is the field's polynomials, with the constant's inverse a coefficient (:polynomial).

Detailed Description

The Rules of Harmony (Groups, Rings, and Fields).

@partition :algebra

Copyright 2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

field__The_Field_of_Reunion

In the tradition of Sharaf al-Dīn al-Ṭūsī, the Field is the species of total inversion. It ensures that every action has a symmetric reaction—a multiplicative inverse—allowing for the 'completion' and 'reunion' of any algebraic equation within its own universe.

We anchor the standard C++ arithmetic operators as formal Algebraic Morphisms within the Dedekind Ontology:

• operator+ : The Additive Group Morphism (The Translation).
• operator* : The Multiplicative Morphism (The Scaling).
• operator- : The Inverse Morphism (The Symmetry).

@build_order 4 @dependency :category, :mereology, :order

See also

dedekind.ontology:category

dedekind.ontology:mereology

dedekind.ontology:order

Wikipedia: Abstract algebra, Group theory, Ring (mathematics), Field (physics)

Copyright

2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

Note

"Es steht alles schon bei Dedekind." ("It is already all in Dedekind.") – Emmy Noether, as quoted by B. L. van der Waerden (1975)