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fix galois_group problem. #4396

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Dec 11, 2024
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50 changes: 40 additions & 10 deletions src/NumberTheory/GaloisGrp/GaloisGrp.jl
Original file line number Diff line number Diff line change
Expand Up @@ -2667,6 +2667,10 @@ function Hecke.absolute_minpoly(a::Oscar.NfNSGenElem{QQFieldElem, QQMPolyRingEle
end

function blow_up(G::PermGroup, C::GaloisCtx, lf::Vector, con::PermGroupElem=one(G))
#TODO: currently con is useless here, the cluster detection and the reduce
# tree is re-arranging the factors in lf randomly
# one would need to trace the re-arranged lf as well
# Since we don't, we have to compute roots and evaluate

if all(x->x[2] == 1, lf)
return G, C
Expand All @@ -2678,28 +2682,53 @@ function blow_up(G::PermGroup, C::GaloisCtx, lf::Vector, con::PermGroupElem=one(

icon = inv(con)

gs = map(Vector{Int}, gens(G))
rr = roots(C, 2; raw = true)
renum = Int[]
for (g, k) = lf
l = findall(iszero, map(g, rr))
append!(renum, l)
end
re = inv(symmetric_group(degree(G))(renum))

gs = map(Vector{Int}, gens(G))

# G is a sub group of prod G_i <= prod sym(n_i) the galois groups
# of the factors
# con describes the current ordering of the roots in so G^con <= prod G_i
#for the factors with multiplicity > 1, we need to append more factors
H = [G]
mps = [gens(G)]
n = degree(G)
for (g,k) = lf
if k == 1
st += degree(g)
continue
end
S = symmetric_group(degree(g))
#need proj G -> G_i, so need a subset of the points (st+1:st+deg(g))
# moved by re, then mapped to 1:deg(g)
h = [S([(i+st)^(gg^re)-st for i=1:degree(g)]) for gg = gens(G)]
for j=2:k
for i=1:degree(g)
mp[n+i] = (st+i)^con
end
for h = gs
for i=1:degree(g)
push!(h, h[(st+i)^con]^icon-st+n)
end
S = symmetric_group(degree(g))
push!(H, S)
push!(mps, h)
for i = 1:degree(g)
mp[n+i] = (st+i)^icon
end
n += degree(g)
end
st += degree(g)
end

D, emb, pro = inner_direct_product(H; morphisms = true)
h = hom(G, D, [prod(emb[i](mps[i][j]) for i=1:length(H)) for j=1:ngens(G)])

C.rt_num = mp
S = symmetric_group(n)
GG, _ = sub(S, map(S, gs))
GG, _ = image(h)

h = hom(G, GG, gens(G), gens(GG))
@assert is_injective(h) && is_surjective(h)
C.G = GG
return GG, C
end

Expand Down Expand Up @@ -2825,6 +2854,7 @@ function galois_group(f::PolyRingElem{<:FieldElem}; prime=0, pStart::Int = 2*deg
@vprint :GaloisGroup 1 "found $(length(cl)) connected components\n"

res = Vector{Tuple{typeof(C[1]), PermGroupElem}}()
llf = Int[]
function setup(C::Vector{<:GaloisCtx})
G, emb, pro = inner_direct_product([x.G for x = C], morphisms = true)
g = prod(x.f for x = C)
Expand Down
15 changes: 13 additions & 2 deletions test/NumberTheory/galthy.jl
Original file line number Diff line number Diff line change
Expand Up @@ -10,7 +10,7 @@
@test degree(L) == order(G)
@test length(roots(L, k.pol)) == 5

R, x = polynomial_ring(QQ, :x)
R, x = polynomial_ring(QQ, :x; cached = false)
pol = x^6 - 366*x^4 - 878*x^3 + 4329*x^2 + 14874*x + 10471
g, C = galois_group(pol)
@test order(g) == 18
Expand All @@ -24,7 +24,7 @@
G, C = galois_group((1//13)*x^2+2)
@test order(G) == 2

K, a = number_field(x^4-2)
K, a = number_field(x^4-2; cached = false)
G, C = galois_group(K)
Gc, Cc = galois_group(K, algorithm = :Complex)
Gs, Cs = galois_group(K, algorithm = :Symbolic)
Expand All @@ -46,6 +46,17 @@
G, C = galois_group((2*x+1)^2)
@test order(G) == 1
@test degree(G) == 2

#from errors:
G, C = galois_group((x^4 + 1)^3 * x^2 * (x^2 - 4*x + 1)^5)
@test order(G) == 8
@test degree(G) == 24
k = fixed_field(C, sub(G, [one(G)])[1])
@test degree(k) == 8

G, C = galois_group((x^3-2)^2*(x^3-5)^2*(x^2-6))
@test order(G) == 36
@test degree(G) == 14
end

import Oscar.GaloisGrp: primitive_by_shape, an_sn_by_shape, cycle_structures
Expand Down
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