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
— Functionfinite_field(p::IntegerUnion, d::Int, s::VarName; cached = true, check = true)
finite_field(q::IntegerUnion, s::VarName; cached = true, check = true)
finite_field(f::FqPolyElem, s::VarName; cached = true, check = true)
Return a tuple $K, a$ consisting of a finite field $K$ of order $q = p^d$ and algebra generator $x$. The string $s$ 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 relative finite field $K = k[X]/(f)$ will be constructed as a finite field with base field $k$.
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)
GF
— FunctionGF(p::IntegerUnion, d::Int, s::String; cached::Bool, check::Bool)
GF(q::IntegerUnion, s::String; cached::Bool, check::Bool)
GF(f::FqPolyRingElem; s::String; cached::Bool, check::Bool)
Return a finite field $K$ of order $q = p^d$. The string $s$ 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 $k$.
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
base_field
— Methodbase_field(F::FqField)
Return the base field of F
.
prime_field
— Methodprime_field(F::FqField)
Return the prime field of F
.
degree
— Methoddegree(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
absolute_degree
— Methodabsolute_degree(a::FqField)
Return the degree of the given finite field over the prime field.
is_absolute
— Methodis_absolute(F::FqField)
Return whether the base field of $F$ is a prime field.
defining_polynomial
— Methoddefining_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
gen
— Methodgen(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.
is_gen
— Methodis_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
.
tr
— Methodtr(x::FqFieldElem)
Return the trace of $x$. This is an element of the base field.
absolute_tr
— Methodabsolute_tr(x::FqFieldElem)
Return the absolute trace of $x$. This is an element of the prime field.
norm
— Methodnorm(x::FqFieldElem)
Return the norm of $x$. This is an element of the base field.
absolute_norm
— Methodabsolute_norm(x::FqFieldElem)
Return the absolute norm of $x$. This is an element of the prime field.
lift
— Methodlift(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
.
lift
— Methodlift(::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