fix galois_group problem. (#4396)

The previous last fix changed the order in which the factors
were processed - this broke unfortunately that part in the end
that adds the multiplicity back in.

The current version is not perfect (unneccessary computations are done)
but fixes the problem

src/NumberTheory/GaloisGrp/GaloisGrp.jl

@@ -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
@@ -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
 
@@ -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)

test/NumberTheory/galthy.jl

@@ -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
@@ -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)
@@ -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