Galois fields

Galois fields

Nemo allows the creation of Galois fields of the form $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$. Note that these are not the same as finite fields of degree 1, as Conway polynomials are not used.

For convenience, the following constructors are provided.

GF(n::UInt)
GF(n::Int)

For example, one can create the Galois field of characteristic $7$ as follows.

R = GF(7)

Elements of the field are then created in the usual way.

a = R(3)

Elements of Galois fields have type gfp_elem, and the type of the parent objects is GaloisField.

The modulus $p$ of an element of a Galois field is stored in its parent object.

The gfp_elem type belong to the abstract type FinFieldElem and the GaloisField parent object type belongs to the abstract type FinField.

Galois field functionality

Galois fields in Nemo implement the residue ring interface of AbstractAlgebra.jl:

https://nemocas.github.io/AbstractAlgebra.jl/residue_rings.html

In addition, all the functionality for generic residue rings is available:

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

Below we describe the functionality that is provided in addition to this interface.

Basic manipulation

characteristic(a::GaloisField) -> fmpz

Return the characteristic of the given Galois field.

order(a::GaloisField) -> fmpz

Return the order, i.e. the number of elements in, the given Galois field.

Examples

F = GF(3)

a = characteristic(F)
b = order(F)