# Integer ring

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

for making Julia `BigInt`

s conform to the AbstractAlgebra.jl Ring interface.

In addition to providing a parent object `ZZ`

for Julia `BigInt`

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

Because `BigInt`

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

s by introducing a type union, called `RingElement`

, which is a union of `AbstractAlgebra.RingElem`

and a number of Julia types, including `BigInt`

. Everywhere that `RingElem`

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

, 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 `BigInt`

type as belonging to `AbstractAlgebra.RingElem`

.

One other technicality is that Julia defines certain functions for `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

Integers have type `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 `Integers{BigInt}`

.

For convenience, we also make `Int`

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

) has type `Integers{Int}`

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

## Integer constructors

In order to construct integers in AbstractAlgebra.jl, one can first construct the integer ring itself. This is accomplished using the following constructor.

`Integers{BigInt}()`

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

representing the ring of integers in AbstractAlgebra.jl.

In practice, one simply uses `ZZ`

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 integer ring and making use of the resulting parent object to coerce various elements into the ring.

**Examples**

```
julia> f = ZZ()
0
julia> g = ZZ(123)
123
julia> h = ZZ(BigInt(1234))
1234
```

## Basic ring functionality

The integer ring in AbstractAlgebra.jl implements the full Ring interface and the Euclidean Ring interface.

We give some examples of such functionality.

**Examples**

```
julia> f = ZZ(12)
12
julia> h = zero(ZZ)
0
julia> k = one(ZZ)
1
julia> isone(k)
true
julia> iszero(f)
false
julia> T = parent(f)
Integers
julia> f == deepcopy(f)
true
julia> g = f + 12
24
julia> h = powmod(f, 12, ZZ(17))
4
julia> flag, q = divides(f, ZZ(3))
(true, 4)
```

## Integer functionality provided by AbstractAlgebra.jl

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

type.

### Basic functionality

`AbstractAlgebra.isunit`

— Method.`isunit(a::Integer)`

Return

`true`

if $a$ is $1$ or $-1$.

**Examples**

```
julia> r = ZZ(-1)
-1
julia> isunit(r)
true
```

### Square root

`AbstractAlgebra.sqrt`

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

Return the integer square root of $a$. If $a$ is not a perfect square an exception is thrown. If

`check`

is set to`false`

this check is not performed.

`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(a::AbstractAlgebra.ResFieldElem{T}) where T <: Integer`

Return

`true`

if $a$ is a square.

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

Return

`true`

if $f$ is a perfect square.

`issquare(a::T) where T <: Integer`

Return true if $a$ is a square.

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

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

`AbstractAlgebra.exp`

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

Return $1$ if $a = 0$, otherwise throw an exception. This function is not generally of use to the user, but is used internally in AbstractAlgebra.jl.

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

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

**Examples**

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

### Coprime bases

`AbstractAlgebra.ppio`

— Method.`ppio(a::T, b::T)`

Split $a$ into $c*d$ where $c = gcd(a, b^\infty)$.

**Examples**

```
julia> c, n = ppio(ZZ(12), ZZ(26))
(4, 3)
```