# 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.

GFMethod
GF(p::T; check::Bool=true) where T <: Integer

Return the finite field $\mathbb{F}_p$, where $p$ is a prime. By default, the integer $p$ is checked with a probabilistic algorithm for primality. When check == false, no check is made, 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

julia> F = GF(13)
Finite field F_13

julia> g = F(3)
3

julia> h = F(g)
3

julia> GF(4)
ERROR: DomainError with 4:
Characteristic is not prime in GF(p)
Stacktrace:
[...]

## Basic field functionality

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

We give some examples of such functionality.

Examples

julia> F = GF(13)
Finite field F_13

julia> f = F(7)
7

julia> h = zero(F)
0

julia> k = one(F)
1

julia> isone(k)
true

julia> iszero(h)
true

julia> T = parent(h)
Finite field F_13

julia> h == deepcopy(h)
true

julia> h = h + 2
2

julia> m = inv(k)
1


## Basic manipulation of fields and elements

dataMethod
data(R::GFElem)

Return the internal data used to represent the finite field element. This coincides with lift except where the internal data ids a machine integer.

liftMethod
data(R::GFElem)

Lift the finite field element to the integers. The result will be a multiprecision integer regardless of how the field element is represented internally.

genMethod
gen(R::GFField{T}) where T <: Integer

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

orderMethod
order(R::GFField)

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

degreeMethod
degree(R::GFField)

Return the degree of the given finite field.

Examples

julia> F = GF(13)
Finite field F_13

julia> d = degree(F)
1

julia> n = order(F)
13

julia> g = gen(F)
1