# 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

`Nemo.finite_field`

— Function```
finite_field(p::IntegerUnion, d::Int, s::VarName = :o; cached::Bool = true, check::Bool = true)
finite_field(q::IntegerUnion, s::VarName = :o; cached::Bool = true, check::Bool = true)
finite_field(f::FqPolyRingElem, s::VarName = :o; cached::Bool = true, check::Bool = true)
```

Return a tuple $(K, x)$ of a finite field $K$ of order $q = p^d$, where $p$ is a prime, and a generator $x$ of $K$ (see `gen`

for a definition). The identifier $s$ is used to designate how the finite field generator will be printed.

If a polynomial $f \in k[X]$ over a finite field $k$ is specified, the finite field $K = k[X]/(f)$ will be constructed as a finite field with base field $k$.

See also `GF`

which only returns $K$.

**Examples**

```
julia> K, a = finite_field(3, 2, "a")
(Finite field of degree 2 and characteristic 3, a)
julia> K, a = finite_field(9, "a")
(Finite field of degree 2 and characteristic 3, a)
julia> Kx, x = K["x"];
julia> L, b = finite_field(x^3 + x^2 + x + 2, "b")
(Finite field of degree 3 over GF(3, 2), b)
```

`Nemo.GF`

— Function```
GF(p::IntegerUnion, d::Int, s::VarName = :o; cached::Bool = true, check::Bool = true)
GF(q::IntegerUnion, s::VarName = :o; cached::Bool = true, check::Bool = true)
GF(f::FqPolyRingElem, s::VarName = :o; cached::Bool = true, check::Bool = true)
```

Return a finite field $K$ of order $q = p^d$, where $p$ is a prime. The identifier $s$ is used to designate how the finite field generator will be printed.

If a polynomial $f \in k[X]$ over a finite field $k$ is specified, the finite field $K = k[X]/(f)$ will be constructed as a finite field with base field $k$.

See also `finite_field`

which additionally returns a finite field generator of $K$.

**Examples**

```
julia> K = GF(3, 2, "a")
Finite field of degree 2 and characteristic 3
julia> K = GF(9, "a")
Finite field of degree 2 and characteristic 3
julia> Kx, x = K["x"];
julia> L = GF(x^3 + x^2 + x + 2, "b")
Finite field of degree 3 over GF(3, 2)
```

## Field functionality

`AbstractAlgebra.Generic.base_field`

— Method`base_field(F::FqField)`

Return the base field of `F`

.

`Nemo.prime_field`

— Method`prime_field(F::FqField)`

Return the prime field of `F`

.

`AbstractAlgebra.degree`

— Method`degree(K::FqField) -> Int`

Return the degree of the given finite field over the base field.

**Examples**

```
julia> K, a = finite_field(3, 2, "a");
julia> degree(K)
2
julia> Kx, x = K["x"];
julia> L, b = finite_field(x^3 + x^2 + x + 2, "b");
julia> degree(L)
3
```

`Nemo.absolute_degree`

— Method`absolute_degree(a::FqField)`

Return the degree of the given finite field over the prime field.

`Nemo.is_absolute`

— Method`is_absolute(F::FqField)`

Return whether the base field of $F$ is a prime field.

`AbstractAlgebra.Generic.defining_polynomial`

— Method`defining_polynomial([R::FqPolyRing], K::FqField)`

Return the defining polynomial of `K`

as a polynomial over the base field of `K`

.

If the polynomial ring `R`

is specified, the polynomial will be an element of `R`

.

**Examples**

```
julia> K, a = finite_field(9, "a");
julia> defining_polynomial(K)
x^2 + 2*x + 2
julia> Ky, y = K["y"];
julia> L, b = finite_field(y^3 + y^2 + y + 2, "b");
julia> defining_polynomial(L)
y^3 + y^2 + y + 2
```

## Element functionality

`AbstractAlgebra.gen`

— Method`gen(L::FqField)`

Return a $K$-algebra generator `a`

of the finite field $L$, where $K$ is the base field of $L$. The element `a`

satisfies `defining_polyomial(a) == 0`

.

Note that this is in general not a multiplicative generator and can be zero, if $L/K$ is an extension of degree one.

`AbstractAlgebra.is_gen`

— Method`is_gen(a::FqFieldElem)`

Return `true`

if the given finite field element is the generator of the finite field over its base field, otherwise return `false`

.

`LinearAlgebra.tr`

— Method`tr(x::FqFieldElem)`

Return the trace of $x$. This is an element of the base field.

`Nemo.absolute_tr`

— Method`absolute_tr(x::FqFieldElem)`

Return the absolute trace of $x$. This is an element of the prime field.

`LinearAlgebra.norm`

— Method`norm(x::FqFieldElem)`

Return the norm of $x$. This is an element of the base field.

`Nemo.absolute_norm`

— Method`absolute_norm(x::FqFieldElem)`

Return the absolute norm of $x$. This is an element of the prime field.

`AbstractAlgebra.lift`

— Method`lift(R::FqPolyRing, a::FqFieldElem) -> FqPolyRingElem`

Given a polynomial ring over the base field of the parent of `a`

, return a lift such that `parent(a)(lift(R, a)) == a`

is `true`

.

`AbstractAlgebra.lift`

— Method`lift(::ZZRing, x::FqFieldElem) -> ZZRingElem`

Given an element $x$ of a prime field $\mathbf{F}_p$, return a preimage under the canonical map $\mathbf{Z} \to \mathbf{F}_p$.

**Examples**

```
julia> K = GF(19);
julia> lift(ZZ, K(3))
3
```