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AlgClosureFp.jl
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AlgClosureFp.jl
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module AlgClosureFp
using Oscar
import Base: +, -, *, //, hash, show, ==
import Oscar: divexact, add!, mul, mul!, addeq!, sub, data
struct AlgClosure{T} <: AbstractAlgebra.Field
# T <: FinField
k::T
fld::Dict{Int, FinField}
function AlgClosure(k::T) where T <: FinField
return new{T}(k, Dict{Int, FinField}(degree(k) => k))
end
end
function show(io::IO, A::AlgClosure)
print(io, "Algebraic Closure of $(A.k)")
end
Oscar.base_field(A::AlgClosure) = A.k
Oscar.base_ring(A::AlgClosure) = A.k
Oscar.characteristic(k::AlgClosure) = characteristic(base_field(k))
struct AlgClosureElem{T} <: FieldElem
# T <: FinFieldElem
data::FinFieldElem
parent::AlgClosure{T}
end
Oscar.elem_type(::AlgClosure{T}) where T = AlgClosureElem{T}
Oscar.parent_type(::AlgClosureElem{T}) where T = AlgClosure{T}
Oscar.parent_type(::Type{AlgClosureElem{T}}) where T = AlgClosure{T}
function show(io::IO, a::AlgClosureElem)
print(io, data(a))
end
function Base.deepcopy_internal(a::AlgClosureElem, d::IdDict)
return AlgClosureElem(data(a), parent(a))
end
(A::AlgClosure)(a::Int) = AlgClosureElem(A.k(a), A)
(A::AlgClosure)(a::AlgClosureElem) = a
(A::AlgClosure)() = A(0)
function (A::AlgClosure)(a::FinFieldElem)
@assert characteristic(parent(a)) == characteristic(A)
if haskey(A.fld, degree(parent(a)))
@assert A.fld[degree(parent(a))] == parent(a)
end
return AlgClosureElem(a, A)
end
Oscar.zero(A::AlgClosure) = AlgClosureElem(zero(base_field(A)), A)
Oscar.one(A::AlgClosure) = AlgClosureElem(one(base_field(A)), A)
Oscar.parent(a::AlgClosureElem) = a.parent
Oscar.data(a::AlgClosureElem) = a.data
function check_parent(a::AlgClosureElem, b::AlgClosureElem)
parent(a) == parent(b) || error("incompatible elements")
end
function ext_of_degree(A::AlgClosure, d::Int)
if haskey(A.fld, d)
return A.fld[d]
end
k = base_ring(A)
if isa(k, Nemo.GaloisField) || isa(k, FqNmodFiniteField)
K = GF(Int(characteristic(k)), d, cached = false)
else
K = GF(characteristic(k), d, cached = false)
end
A.fld[d] = K
return K
end
function op(f::Function, a::AlgClosureElem, b::AlgClosureElem)
check_parent(a, b)
ad = data(a)
bd = data(b)
if parent(ad) == parent(bd)
return f(ad,bd)
end
l = lcm(degree(parent(ad)), degree(parent(bd)))
k = ext_of_degree(parent(a), l)
embed(parent(ad), k)
embed(parent(bd), k)
return f(k(ad), k(bd))
end
function Oscar.embed(k::Nemo.GaloisField, K::FqNmodFiniteField)
@assert characteristic(K) == characteristic(k)
end
+(a::AlgClosureElem, b::AlgClosureElem) = AlgClosureElem(op(+, a, b), parent(a))
-(a::AlgClosureElem, b::AlgClosureElem) = AlgClosureElem(op(-, a, b), parent(a))
*(a::AlgClosureElem{T}, b::AlgClosureElem{S}) where S where T = AlgClosureElem(op(*, a, b), parent(a))
//(a::AlgClosureElem, b::AlgClosureElem) = AlgClosureElem(op(//, a, b), parent(a))
divexact(a::AlgClosureElem, b::AlgClosureElem) = AlgClosureElem(op(divexact, a, b), parent(a))
==(a::AlgClosureElem, b::AlgClosureElem) = op(==, a, b)
Oscar.iszero(a::AlgClosureElem) = iszero(data(a))
Oscar.isone(a::AlgClosureElem) = isone(data(a))
-(a::AlgClosureElem) = AlgClosureElem(-data(a), parent(a))
function Oscar.roots(a::AlgClosureElem, b::Int)
ad = data(a)
kx, x = PolynomialRing(parent(ad), cached = false)
f = x^b-ad
lf = factor(f)
d = mapreduce(degree, lcm, keys(lf.fac), init = 1)
d = lcm(d, degree(parent(ad)))
K = ext_of_degree(parent(a), d)
r = roots(f, K)
return [AlgClosureElem(x, parent(a)) for x = r]
end
function Oscar.roots(a::Generic.Poly{AlgClosureElem{T}}) where T
A = base_ring(a)
b = minimize(FinField, collect(coefficients(a)))
kx, x = PolynomialRing(parent(b[1]), cached = false)
f = kx(b)
lf = factor(f)
d = mapreduce(degree, lcm, keys(lf.fac), init = 1)
d = lcm(d, degree(parent(b[1])))
K = ext_of_degree(A, d)
r = roots(f, K)
return [AlgClosureElem(x, A) for x = r]
end
function Oscar.minpoly(a::AlgClosureElem)
return minpoly(data(a))
end
function Oscar.minpoly(a::gfp_elem)
kx, x = PolynomialRing(parent(a), cached = false)
return x-a
end
function Oscar.degree(a::AlgClosureElem)
#TODO: via Frobenius? as a fixed s.th.?
return degree(minpoly(data(a)))
end
function minimize(a::AlgClosureElem)
f = minpoly(a)
k = ext_of_degree(parent(a), degree(f))
embed(k, parent(data(a)))
return AlgClosureElem(k(data(a)), parent(a))
end
function minimize(::Type{FinField}, a::AlgClosureElem)
return data(minimize(a))
end
function minimize(::Type{FinField}, a::AbstractArray{<:AlgClosureElem})
if length(a) == 0
return a
end
@assert all(x->parent(x) == parent(a[1]), a)
da = map(degree, a)
l = reduce(lcm, da)
k = ext_of_degree(parent(a[1]), l)
b = elem_type(k)[]
for i = eachindex(a)
if da[i] < l
embed(parent(data(a[i])), k)
push!(b, k(data(a[i])))
elseif da[i] == l
push!(b, data(a[i]))
else
embed(k, parent(data(a[i])))
push!(b, k(data(a[i])))
end
end
return b
end
function (F::FinField)(a::AlgClosureElem)
b = minimize(FinField, a)
embed(parent(b), F)
return F(b)
end
function Base.hash(a::AlgClosureElem, u::UInt)
b = minimize(a)
return hash(data(b), u)
end
function Oscar.gmodule(::Type{FinField}, C::GModule{<:Any, <:Generic.FreeModule{<:AlgClosureElem{<:FinField}}})
d = dim(C)
l = 1
for g = C.ac
l = lcm(l, lcm(collect(map_entries(x->Hecke.degree(parent(x.data)), mat(g)))))
end
K = ext_of_degree(base_ring(C), l)
return gmodule(K, C)
end
function Oscar.map_entries(K::FinField, M::MatElem{<:AlgClosureElem})
N = zero_matrix(K, nrows(M), ncols(M))
for i=1:nrows(M)
for j=1:ncols(M)
embed(parent(data(M[i,j])), K)
N[i,j] = K(data(M[i,j]))
end
end
return N
end
function Oscar.gmodule(K::FinField, C::GModule{<:Any, <:Generic.FreeModule{<:AlgClosureElem{<:FinField}}})
d = dim(C)
F = free_module(K, d)
if d == 0
h = hom(F, F, elem_type(F)[])
return gmodule(F, group(C), typeof(h)[hom(F, F, map_entries(K, mat(x))) for x = C.ac])
end
return gmodule(F, group(C), [hom(F, F, map_entries(K, mat(x))) for x = C.ac])
end
function Oscar.gmodule(K::FinField, C::GModule{<:Any, <:Generic.FreeModule{<:FinFieldElem}})
d = dim(C)
F = free_module(K, d)
if d == 0
h = hom(F, F, elem_type(F)[])
return gmodule(F, group(C), typeof(h)[hom(F, F, map_entries(K, mat(x))) for x = C.ac])
end
return gmodule(F, group(C), [hom(F, F, map_entries(K, mat(x))) for x = C.ac])
end
function Oscar.GModuleFromGap.hom_base(C::T, D::T) where T <: GModule{<:Any, <:Generic.FreeModule{<:AlgClosureElem{<:FinField}}}
C1 = gmodule(FinField, C)
D1 = gmodule(FinField, D)
Cf = degree(base_ring(C1))
Df = degree(base_ring(D1))
l = lcm(Cf, Df)
K = ext_of_degree(base_ring(C), l)
if l != Cf
C1 = gmodule(K, C1)
end
if l != Df
D1 = gmodule(K, D1)
end
h = Oscar.GModuleFromGap.hom_base(C1, D1)
if length(h) == 0
return h
end
return map(x->map_entries(base_ring(C), x), h)
end
end # AlgClosureFp