Integers mod p

Integers mod a prime $p$ are implemented via the Singular n_Zp type for any positive prime modulus less than $2^{29}$.

The associated field of integers mod $p$ is represented by a parent object which can be constructed by a call to the Fp constructor.

The types of the parent objects and elements of the associated fields of integers modulo $p$ are given in the following table according to the library providing them.

All integer mod $p$ element types belong directly to the abstract type FieldElem and all the parent object types belong to the abstract type Field.

Integer mod $p$ functionality

Singular.jl integers modulo $p$ provides the field and residue ring functionality of AbstractAlgebra.

https://nemocas.github.io/AbstractAlgebra.jl/latest/field

https://nemocas.github.io/AbstractAlgebra.jl/latest/residue

Below, we describe the functionality that is specific to the Singular integers mod $p$ field and not already listed at the given links.

Constructors

The following constructors are available to create the field of integers modulo a prime $p$.

Fp(p::Int; cached=true)

Construct the field of integers modulo $p$. By default, the field is cached, so that all fields of integers modulo $p$ have the same parent object. If this is not the desired behaviour, the cached parameter can be set to false. If $p$ is not a prime or $p$ is not in the range $(0, 2^{29})$, an exception is raised.

Given a field $R$ of integers modulo $p$, we also have the following coercions in addition to the standard ones expected.

R(n::n_Z)
R(n::ZZRingElem)

Coerce a Singular or Flint integer value into the field.

Basic manipulation

Examples

julia> R = Fp(23)
Finite field of characteristic 23

julia> a = R(5)
5

julia> is_unit(a)
true

julia> c = characteristic(R)
23

Conversions

Int(n::n_Zp)

Lift the integer $n$ modulo $p$ to a Julia Int. The result is always in the range $[0, p)$.

Examples

julia> R = Fp(23)
Finite field of characteristic 23

julia> a = R(5)
5

julia> b = Int(a)
5