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Finite fields · AbstractAlgebra.jl
julia
https://nemocas.github.io/AbstractAlgebra.jl/latest/finfield/
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3 月前
AbstractAlgebra.jl
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AbstractAlgebra.jl
Fundamental interface of AbstractAlgebra.jl
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FieldsFinite fieldsFinite fieldsGitHubFinite fieldsAbstractAlgebra.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 objectsFinite fields have type GFField{T} where T is either Int or BigInt.Elements of such a finite field have type GFElem{T}.Finite field constructorsIn 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 <: IntegerReturn 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.sourceHere are some examples of creating a finite field and making use of the resulting parent object to coerce various elements into the field.Examplesjulia> F = GF(13) Finite field F_13 julia> g = F(3) julia> h = F(g) julia> GF(4) ERROR: DomainError with 4: Characteristic is not prime in GF(p) Stacktrace: [...]Basic field functionalityThe finite field module in AbstractAlgebra.jl implements the full Field interface.We give some examples of such functionality.Examplesjulia> F = GF(13) Finite field F_13 julia> f = F(7) julia> h = zero(F) julia> k = one(F) julia> isone(k) julia> iszero(h) julia> T = parent(h) Finite field F_13 julia> h == deepcopy(h) julia> h = h + 2 julia> m = inv(k) Basic manipulation of fields and elementsAbstractAlgebra.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.sourceAbstractAlgebra.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.sourceAbstractAlgebra.gen — Methodgen(R::GFField{T}) where T <: IntegerReturn a generator of the field. Currently this returns 1.sourceAbstractAlgebra.order — Methodorder(R::GFField)Return the order, i.e. the number of element in the given finite field.sourceAbstractAlgebra.degree — Methoddegree(R::GFField)Return the degree of the given finite field.sourceExamplesjulia> F = GF(13) Finite field F_13 julia> d = degree(F)
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