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(p::T) where T <: Integer

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

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)