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
— MethodGF(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
AbstractAlgebra.data
— Methoddata(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.
AbstractAlgebra.lift
— Methodlift(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.
AbstractAlgebra.gen
— Methodgen(R::GFField{T}) where T <: Integer
Return a generator of the field. Currently this returns 1.
AbstractAlgebra.order
— Methodorder(R::GFField)
Return the order, i.e. the number of element in the given finite field.
AbstractAlgebra.degree
— Methoddegree(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