Add some helpers for creation of embeddings (#1296)

Author: thofma

Project.toml

@@ -1,6 +1,6 @@
 name = "Hecke"
 uuid = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
-version = "0.22.8"
+version = "0.22.9"
 
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"

src/NumField/ComplexEmbeddings/NfRel.jl

@@ -252,3 +252,27 @@ function restrict(e::NumFieldEmb, f::NumFieldMor{<: NfRel, <: Any, <: Any})
   i = findfirst(cn)
   return emb[i]
 end
+
+################################################################################
+#
+#  Creation of complex embedding
+#
+################################################################################
+
+function complex_embedding(K::NfRel, e::NumFieldEmb, r::acb)
+  @req number_field(e) === base_field(K) "Embedding must be embedding of base field"
+  embs = complex_embeddings(K)
+  cnt = 0
+  local eee::embedding_type(K)
+  for ee in embs
+    if ee.base_field_emb != e
+      continue
+    end
+    if overlaps(ee.r, r)
+      eee = ee
+      cnt += 1
+    end
+  end
+  cnt != 1 && error("Something wrong")
+  return eee
+end

src/NumField/ComplexEmbeddings/NfRelNS.jl

@@ -228,3 +228,27 @@ function restrict(e::NumFieldEmb, f::NumFieldMor{<: NfRelNS, <: Any, <: Any})
   i = findfirst(cn)
   return emb[i]
 end
+
+################################################################################
+#
+#  Creation of complex embedding
+#
+################################################################################
+
+function complex_embedding(K::NfRelNS, e::NumFieldEmb, r::Vector{acb})
+  @req number_field(e) === base_field(K) "Embedding must be embedding of base field"
+  embs = complex_embeddings(K)
+  cnt = 0
+  local eee::embedding_type(K)
+  for ee in embs
+    if ee.base_field_emb != e
+      continue
+    end
+    if all(x -> overlaps(x[1], x[2]), zip(ee.data, r))
+      eee = ee
+      cnt += 1
+    end
+  end
+  cnt != 1 && error("Something wrong")
+  return eee
+end

test/NumField/ComplexEmbeddings/NfRel.jl

@@ -98,4 +98,12 @@
   @test Set([restrict(e, L) for e in emb]) == Set(complex_embeddings(L))
   @test Set([restrict(e, K) for e in emb]) == Set(complex_embeddings(K))
 
+  # Creation
+
+  Qx, x = QQ["x"]
+  K, a = number_field(x^2 - 2, "a")
+  L, b = number_field(polynomial(K, [-3, 0, 1], cached = false))
+  for e in complex_embeddings(L)
+    @test complex_embedding(L, restrict(e, K), e(b)) == e
+  end
 end

test/NumField/ComplexEmbeddings/NfRelNS.jl

@@ -123,4 +123,13 @@
   emb = complex_embeddings(M)
   @test Set([restrict(e, L) for e in emb]) == Set(complex_embeddings(L))
   @test Set([restrict(e, K) for e in emb]) == Set(complex_embeddings(K))
+
+  # Creation
+
+  Qx, x = QQ["x"]
+  K, a = number_field(x^2 - 2, "a")
+  L, b = number_field([polynomial(K, [-3, 0, 1], cached = false)])
+  for e in complex_embeddings(L)
+    @test complex_embedding(L, restrict(e, K), e.(b)) == e
+  end
 end