# 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 and no generator is given.

For convenience, the following constructors are provided.

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

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 when $p$ is given to the constructor as an Int or UInt, and of type gfp_fmpz_elem if $p$ is given as an fmpz, and the type of the parent objects is GaloisField or GaloisFmpzField respectively.

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

The gfp_elem and gfp_fmpz_elem types belong to the abstract type FinFieldElem and the GaloisField and GaloisFmpzField parent object types belong to the abstract type FinField.

## Galois field functionality

Galois fields in Nemo provide all the residue ring functionality of AbstractAlgebra.jl:

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

In addition, all the functionality for rings is available:

https://nemocas.github.io/AbstractAlgebra.jl/stable/ring

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

## Basic manipulation

Examples

F = GF(3)

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