finitefield.md

CurrentModule = Nemo
DocTestSetup = quote
    using Nemo
end

Finite fields

A finite field $K$ is represented as simple extension $K = k(\alpha) = k[x]/(f)$, where $k$ can be

• a prime field $\mathbf{F}_p$ ($K$ is then an absolute finite field), or
• an arbitrary finite field $k$ ($K$ is then a relative finite field).

In both cases, we call $k$ the base field of $K$, $\alpha$ a generator and $f$ the defining polynomial of $K$.

Note that all field theoretic properties (like basis, degree or trace) are defined with respect to the base field. Methods with prefix absolute_ return

Finite field functionality

Finite fields in Nemo provide all the field functionality described in AbstractAlgebra:

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

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

Constructors

finite_field
GF

Field functionality

base_field(::FqField)
prime_field(::FqField)
degree(::FqField)
absolute_degree(::FqField)
is_absolute(::FqField)
defining_polynomial(::FqPolyRing, ::FqField)

Element functionality

gen(::FqField)
is_gen(::FqFieldElem)
tr(::FqFieldElem)
absolute_tr(::FqFieldElem)
norm(::FqFieldElem)
absolute_norm(::FqFieldElem)
lift(::FqPolyRing, ::FqFieldElem)
lift(::ZZRing, ::FqFieldElem)