feat: extend methods for complex_conjugation (#1426)

- Allow arbitrary NumField
- Allow totally real fields
- Allow QQ

Project.toml

@@ -1,6 +1,6 @@
 name = "Hecke"
 uuid = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
-version = "0.30.1"
+version = "0.30.2"
 
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"

src/NumField/ComplexEmbeddings/Generic.jl

@@ -488,3 +488,16 @@ function _log_evaluate_fac_elem(e, x, prec)
   end
   return z
 end
+
+################################################################################
+#
+#  Complex conjugation
+#
+################################################################################
+
+function complex_conjugation(K::NumField)
+  L, f = absolute_simple_field(K)
+  g = inv(f)
+  conj = complex_conjugation(L)
+  return compose(compose(g, conj), f)
+end

src/NumField/ComplexEmbeddings/QQ.jl

@@ -52,3 +52,5 @@ sign(x::Union{ZZRingElem, FacElem{QQFieldElem}}, ::QQEmb) = sign(Int, x)
 signs(x::Union{ZZRingElem, QQFieldElem, FacElem{QQFieldElem}}, ::Vector{QQEmb}) = Dict(QQEmb() => sign(x, QQEmb()))
 
 signs(x::Union{ZZRingElem, QQFieldElem, FacElem{QQFieldElem}}) = Dict(QQEmb() => sign(x, QQEmb()))
+
+complex_conjugation(K::QQField) = identity_map(K)

src/NumField/NfAbs/Conjugates.jl

@@ -491,6 +491,9 @@ Given a totally complex normal number field, this function returns an
 automorphism which is the restriction of complex conjugation at one embedding.
 """
 function complex_conjugation(K::AbsSimpleNumField; auts::Vector{<:NumFieldHom{AbsSimpleNumField, AbsSimpleNumField}} = morphism_type(AbsSimpleNumField, AbsSimpleNumField)[])
+  if is_totally_real(K)
+    return id_hom(K)
+  end
   if !isempty(auts)
     A = auts
   else

test/NfAbs/Conjugates.jl

@@ -110,6 +110,9 @@ end
   K, a = cyclotomic_field(13)
   aut = @inferred complex_conjugation(K)
   @test aut(a) == a^-1
+  K, a = quadratic_field(5)
+  aut = @inferred complex_conjugation(K)
+  @test aut(a) == a
 end
 
 @testset "Bad example" begin

test/NumField/ComplexEmbeddings/Generic.jl

@@ -61,6 +61,11 @@
 
     c = FacElem(K(2))^1000 * FacElem(K(3))^-1000 * FacElem(K(5))^1000
     @test (@inferred e(c)) isa AcbFieldElem
+
+    c = complex_conjugation(K)
+    if !(K isa QQField)
+      @test c == id_hom(K)
+    end
   end
 
   K, a = quadratic_field(-1)
@@ -73,4 +78,23 @@
   e = complex_embeddings(k)[1]
   eext = extend(e, ktoK)
   @test all(overlaps(c(ktoK(b)), e(b)) for c in eext)
+
+  Qx, x = QQ["x"]
+  K, _ = rationals_as_number_field()
+  Kt, t = K["t"]
+  K1, = number_field(t^2 - 2)
+  K2, = number_field([t^2 - 2, t^2 - 3])
+  K3, = number_field([x^2 - 2, x^2 - 3])
+  for K in Any[K1, K2, K3]
+    c = complex_conjugation(K)
+    @test c == id_hom(K)
+  end
+
+  K1, = number_field(t^2 + 2)
+  K2, = number_field([t^2 + 2])
+  K3, = number_field([x^2 + 2])
+  for K in Any[K1, K2, K3]
+    c = complex_conjugation(K)
+    @test c != id_hom(K)
+  end
 end

test/NumField/ComplexEmbeddings/QQ.jl

@@ -1,2 +1,4 @@
 @testset "QQ" begin
+  c = complex_conjugation(QQ)
+  @test c(QQ(1)) == QQ(1)
 end