# Finite fields

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

for finite fields. The module is a naive implementation that supports only fields of degree $1$ (prime fields). They are modelled as $\mathbb{Z}/p\mathbb{Z}$ for $p$ a prime.

## Types and parent objects

Finite fields have type `GFField{T}`

where `T`

is either `Int`

or `BigInt`

.

Elements of such a finite field have type `gfelem{T}`

.

## Finite field constructors

In order to construct finite fields in AbstractAlgebra.jl, one must first construct the field itself. This is accomplished with the following constructors.

`AbstractAlgebra.GF`

— Method.GF{T <: Integer}(p::T)

Return the finite field $\mathbb{F}_p$, where $p$ is a prime. The integer $p$ is not checked for primality, but the behaviour of the resulting object is undefined if $p$ is composite.

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

**Examples**

```
F = GF(13)
g = F(3)
h = F(g)
```

## Basic field functionality

The finite field module in AbstractAlgebra.jl implements the full Field interface.

We give some examples of such functionality.

**Examples**

```
F = GF(13)
h = zero(F)
k = one(F)
isone(k) == true
iszero(f) == false
U = base_ring(F)
V = base_ring(h)
T = parent(h)
h == deepcopy(h)
h = h + 2
m = inv(k)
```

## Basic manipulation of fields and elements

`AbstractAlgebra.Generic.gen`

— Method.`gen{T <: Integer}(a::GFField{T})`

Return a generator of the field. Currently this returns 1.

`AbstractAlgebra.Generic.order`

— Method.`order(R::GFField)`

Return the order, i.e. the number of element in, the given finite field.

`AbstractAlgebra.Generic.degree`

— Method.`degree(R::GFField)`

Return the degree of the given finite field.

**Examples**

```
F = GF(13)
d = degree(F)
n = order(F)
g = gen(F)
```