Fraction Field Interface

Fraction Field Interface

Fraction fields are supported in AbstractAlgebra.jl, at least for gcd domains. In addition to the standard Ring interface, some additional functions are required to be present for fraction fields.

Types and parents

AbstractAlgebra provides two abstract types for fraction fields and their elements:

We have that FracField{T} <: AbstractAlgebra.Field and FracElem{T} <: AbstractAlgebra.FieldElem.

Note that both abstract types are parameterised. The type T should usually be the type of elements of the base ring of the fraction field.

Fraction fields should be made unique on the system by caching parent objects (unless an optional cache parameter is set to false). Fraction fields should at least be distinguished based on their base ring.

See src/generic/GenericTypes.jl for an example of how to implement such a cache (which usually makes use of a dictionary).

Required functionality for fraction fields

In addition to the required functionality for the Field interface the Fraction Field interface has the following required functions.

We suppose that R is a fictitious base ring, and that S is the fraction field with parent object S of type MyFracField{T}. We also assume the fractions in the field have type MyFrac{T}, where T is the type of elements of the base ring.

Of course, in practice these types may not be parameterised, but we use parameterised types here to make the interface clearer.

Note that the type T must (transitively) belong to the abstract type RingElem.

Constructors

We provide the following constructors. Note that these constructors don't require construction of the parent object first. This is easier to achieve if the fraction element type doesn't contain a reference to the parent object, but merely contains a reference to the base ring. The parent object can then be constructed on demand.

//(x::T, y::T) where T <: AbstractAlgebra.RingElem

Return the fraction $x/y$.

//(x::T, y::AbstractAlgebra.FracElem{T}) where T <: AbstractAlgebra.RingElem

Return $x/y$ where $x$ is in the base ring of $y$.

//(x::AbstractAlgebra.FracElem{T}, y::T) where T <: AbstractAlgebra.RingElem

Return $x/y$ where $y$ is in the base ring of $x$.

Examples

R, x = PolynomialRing(ZZ, "x")

f = (x^2 + x + 1)//(x^3 + 3x + 1)
g = f//x
h = x//f

Basic manipulation of fields and elements

numerator(d::MyFrac{T}) where T <: AbstractAlgebra.RingElem

Given a fraction $d = a/b$ return $a$, where $a/b$ is in lowest terms with respect to the canonical_unit and gcd functions on the base ring.

denominator(d::MyFrac{T}) where T <: AbstractAlgebra.RingElem

Given a fraction $d = a/b$ return $b$, where $a/b$ is in lowest terms with respect to the canonical_unit and gcd functions on the base ring.

Examples

R, x = PolynomialRing(QQ, "x")

f = (x^2 + x + 1)//(x^3 + 3x + 1)

n = numerator(f)
d = denominator(f)