Merge 6ce533e into 6b87cbe

src/Misc.jl

@@ -35,3 +35,4 @@ include("Misc/RelFiniteField.jl")
 include("Misc/Plesken.jl")
 include("Misc/power_residue_symbol.jl")
 include("Misc/AVLTrees.jl")
+include("Misc/QQBar.jl")

src/Misc/QQBar.jl

@@ -0,0 +1,66 @@
+function _primitive_element(a::Vector{QQBarFieldElem})
+  pe = a[1]
+  f = minpoly(Globals.Qx, pe)
+  Qx = parent(f)
+  local k, f
+  for i = 2:length(a)
+    g = minpoly(Globals.Qx, a[i])
+    f = minpoly(Globals.Qx, pe)
+    k, _ = number_field(f, check = false, cached = false)
+    lf = collect(keys(factor(k, g).fac))
+    for j = 1:length(lf)
+      h = map_coefficients(x->Qx(x)(pe), lf[j])
+      if is_zero(h(a[i]))
+        d = degree(f) * degree(h)
+        mu = 0
+        while degree(minpoly(Globals.Qx, pe+mu*a[i])) != d
+          mu += 1
+          if mu > 10
+            error("too bad")
+          end
+        end
+        pe += mu*a[i]
+      end
+    end
+  end
+  return pe
+end
+
+function _map_to_common_number_field(a::Vector{QQBarFieldElem})
+  res = Vector{SimpleNumFieldElem}(undef, length(a))
+  if length(a) == 0
+    return res
+  end
+  pe = _primitive_element(a)
+  f = minpoly(Globals.Qx, pe)
+  k, = number_field(f; cached = false, check = false )
+  f = minpoly(Globals.Qx, pe)
+  for i in 1:length(a)
+    g = minpoly(Globals.Qx, a[i])
+    rts = roots(k, g)
+    for r in rts
+      if evaluate(Globals.Qx(r), pe) == a[i]
+        res[i] = r
+        break
+      end
+    end
+  end
+  return res, pe
+end
+
+# TODO: use _qqbar_roots_poly_squarefree as soon as available in FLINT_jll
+
+function factor(f::PolyRingElem{QQBarFieldElem})
+  # the roots of f in Qbar are contained in the roots of norm(f) in Qbar
+  @req !is_zero(f) "Polynomial must be non-zero"
+  QQbar = base_ring(f)
+  cfs, pe = _map_to_common_number_field(collect(coefficients(f)))
+  K = parent(cfs[1])
+  Kt, t = polynomial_ring(K; cached = false)
+  fK = Kt(cfs)
+  rts = roots(QQbar, norm(fK))
+  facts = [gen(parent(f)) - r for r in rts if is_zero(f(r))]
+  fac = Fac(parent(f)(leading_coefficient(f)), Dict(facts[i] => valuation(f, facts[i]) for i in 1:length(facts)))
+  @assert degree(f) == sum(e for (_,e) in fac)
+  return fac
+end

test/Misc.jl

@@ -15,4 +15,5 @@
   include("Misc/PIDIdeal.jl")
   include("Misc/MSet.jl")
   include("Misc/AVLTrees.jl")
+  include("Misc/QQBar.jl")
 end

test/Misc/QQBar.jl

@@ -0,0 +1,8 @@
+let
+  K = algebraic_closure(QQ)
+  Kx, x = polynomial_ring(K; cached = false)
+  f = 2*(x^3 + sqrt(K(2)))^2
+  fa = factor(f)
+  @test f == unit(fa) * prod(g^e for (g, e) in fa)
+  @test all(degree(g) == 1 for (g, e) in fa)
+end