# Rational field

AbstractAlgebra.jl provides a module, implemented in `src/julia/Rational.jl`

for making Julia `Rational{BigInt}`

s conform to the AbstractAlgebra.jl Field interface.

In addition to providing a parent object `QQ`

for Julia `Rational{BigInt}`

s, we implement any additional functionality required by AbstractAlgebra.jl.

Because `Rational{BigInt}`

cannot be directly included in the AbstractAlgebra.jl abstract type hierarchy, we achieve integration of Julia `Rational{BigInt}`

s by introducing a type union, called `FieldElement`

, which is a union of `AbstractAlgebra.FieldElem`

and a number of Julia types, including `Rational{BigInt}`

. Everywhere that `FieldElem`

is notionally used in AbstractAlgebra.jl, we are in fact using `FieldElement`

, with additional care being taken to avoid ambiguities.

The details of how this is done are technical, and we refer the reader to the implementation for details. For most intents and purposes, one can think of the Julia `Rational{BigInt}`

type as belonging to `AbstractAlgebra.FieldElem`

.

One other technicality is that Julia defines certain functions for `Rational{BigInt}`

, such as `sqrt`

and `exp`

differently to what AbstractAlgebra.jl requires. To get around this, we redefine these functions internally to AbstractAlgebra.jl, without redefining them for users of AbstractAlgebra.jl. This allows the internals of AbstractAlgebra.jl to function correctly, without broadcasting pirate definitions of already defined Julia functions to the world.

To access the internal definitions, one can use `AbstractAlgebra.sqrt`

and `AbstractAlgebra.exp`

, etc.

## Types and parent objects

Rationals have type `Rational{BigInt}`

, as in Julia itself. We simply supplement the functionality for this type as required for computer algebra.

The parent objects of such integers has type `Rationals{BigInt}`

.

For convenience, we also make `Rational{Int}`

a part of the AbstractAlgebra.jl type hierarchy and its parent object (accessible as `qq`

) has type `Rationals{Int}`

. But we caution that this type is not particularly useful as a model of the rationals and may not function as expected within AbstractAlgebra.jl.

## Rational constructors

In order to construct rationals in AbstractAlgebra.jl, one can first construct the rational field itself. This is accomplished using either of the following constructors.

`FractionField(R::Integers{BigInt})`

`Rationals{BigInt}()`

This gives the unique object of type `Rationals{BigInt}`

representing the field of rationals in AbstractAlgebra.jl.

In practice, one simply uses `QQ`

which is assigned to be the return value of the above constructor. There is no need to call the constructor in practice.

Here are some examples of creating the rational field and making use of the resulting parent object to coerce various elements into the field.

**Examples**

```
julia> f = QQ()
0//1
julia> g = QQ(123)
123//1
julia> h = QQ(BigInt(1234))
1234//1
julia> k = QQ(BigInt(12), BigInt(7))
12//7
julia> QQ == FractionField(ZZ)
true
```

## Basic field functionality

The rational field in AbstractAlgebra.jl implements the full Field and Fraction Field interfaces.

We give some examples of such functionality.

**Examples**

```
julia> f = QQ(12, 7)
12//7
julia> h = zero(QQ)
0//1
julia> k = one(QQ)
1//1
julia> isone(k)
true
julia> iszero(f)
false
julia> U = base_ring(QQ)
Integers
julia> V = base_ring(f)
Integers
julia> T = parent(f)
Rationals
julia> f == deepcopy(f)
true
julia> g = f + 12
96//7
julia> r = ZZ(12)//ZZ(7)
12//7
julia> n = numerator(r)
12
```

## Rational functionality provided by AbstractAlgebra.jl

The functionality below supplements that provided by Julia itself for its `Rational{BigInt}`

type.

### Square root

`AbstractAlgebra.sqrt`

— Method.`sqrt(a::Rational{T}) where T <: Integer`

Return the square root of $a$ if it is the square of a rational, otherwise throw an error.

`AbstractAlgebra.Generic.issquare`

— Method.`issquare(f::AbstractAlgebra.PolyElem{T}) where T <: RingElement`

Return

`true`

if $f$ is a perfect square.

`issquare(a::Rational{T}) where T <: Integer`

Return true if $a$ is the square of a rational.

`AbstractAlgebra.exp`

— Method.`exp(a::Rational{T}) where T <: Integer`

Return $1$ if $a = 0$, otherwise throw an exception.

**Examples**

```
julia> d = AbstractAlgebra.sqrt(ZZ(36)//ZZ(25))
6//5
julia> issquare(ZZ(9)//ZZ(16))
true
julia> m = AbstractAlgebra.exp(ZZ(0)//ZZ(1))
1//1
```